As the question is framed, I don't see why Hinduism needs to refute Islamic beliefs about its own scripture. None of Hinduism's tenets are affected by Islam's tenets.
The question of refutation comes when two or more opposing parties are contending the same issue. Such as when Advaitins, Vishishtadvaitins and Dvaitins all subscribe to the authority of the Vedas, Brahmasutra and Gita, but have differing interpretations of the statements in these scriptures.
The scriptures of Hinduism and Islam and mutually exclusive. So logically speaking, refutation does not make any sense.
However, from a universal point of view of comparative logic, you have trapped yourself in a situation from which you cannot come out. You have started with a definition of Apaurusheya which anyone can claim for any scripture, and which will liberate them from any burden of evidence. So no refutation is possible on any grounds.
Equally, from this universal logical point of view, no scripture can claim to be "uncreated" or "apaurusheya" in its literal form. Language is a human invention, and hence any scripture, which is a fixed collection of words, is a human invention in space and time. So every scripture can be dated and located.
So what else is a valid definition of "apaurusheya"?
In this case, if we examine the Vedas and allied scriptures themselves rather than tiptoeing around them and blindly engaging in superstitious myth-making, then we will be in a completely victorious position.
Here are some examples of the words in the Vedas themselves:
- RV 9.112.3: kārurahaṃ tato bhiṣagupalaprakṣiṇī nanā — “I am a maker of mantras, my father (or son) is a doctor, my mother (or sister) is a flour-maker.”
- RV 1.61.4: asmā idu stomaṃ saṃ hinomi rathaṃ na taṣṭeva — “For him, I design this hymn, just as a carpenter designs a chariot”
- RV 1.77.1: kathā dāśemāgnaye kāsmai devajuṣṭocyate bhāmine gīḥ — “ How shall we offer to Agni, what kind of words should we say to him the bright one, which will be divinely acceptable?”
- RV 1.94.1: imaṃ stomamarhate jātavedase rathamiva saṃ mahemā manīṣayā — “This stotra we make for the most sacred Jātaveda with deep meditation, just like building a chariot”
- RV 1.143.1: pra tavyasīṃ navyasīṃ dhītimagnaye vāco matiṃ sahasaḥ sūnave bhare — “I offer this newest and most expansive ritual and words to the Son of Strength”
- RV 2.19.8: evā te gṛtsamadāḥ śūra manmāvasyavo na vayunāni takṣuḥ — “Thus, we Gṛtsamadas have designed these words for you, O Indra, just as a pathfinder makes paths”
There are hundreds of such statements in the Rig Veda mantras. The rishi is very conscious of making this extraordinary effort with skill and meditation, to create or design a beautiful hymn to offer to the deity. And they want the deity to know that the hymn is coming from them as a gift.
Similarly, Yāskācārya says in his Niruktam (7.3):
एवमुच्चावचैरभिप्रायैः ऋषीणां मन्त्रदृष्टयो भवन्ति
evamuccāvacairabhiprāyaiḥ ṛṣīṇāṃ mantradṛṣṭayo bhavanti
Thus, the visions of the rishis occur with various intentions and topics.
This shows that the ancient tradition knew the Vedas not as a monolithic "golden book" dropped from heaven or from God's mouth or "existing uncreated" in a literal sense.
The Niruktam (1.20) also says:
साक्षात्कृतधर्माण ऋषयो बभूवुः । तेऽवरेभ्योऽसाक्षात्कृतधर्मभ्य उपदेशेन मन्त्रान् संप्रादुः । … वेदं च वेदाङ्गानि च ।
The rishis were the ones who had a direct vision of the truth. For the sake of the inferior people who were not endowed with the vision, they gave these mantras as instruction/teaching. This is the Veda and Vedāṅgas.
Now this introduces a more authentic and verifiable definition of "apaurusheya". The focus is on the eternal truth that is apaurusheya. Not the words and texts, but the eternal truth that they are conveying. The rishis who experienced the truth are teaching others the same truth through the Vedas, so that others may experience the same truth.
The Vedas are unique because the rishis have divine vision. Through intense tapas, they have become enlightened, they have achieved brahmasākṣātkāra, i.e. realization of the Ultimate Reality or Truth. This enlightenment occurs at a level beyond the mind and speech, where the rishis lose their individuality and experience the highest saccidānanda (absolute existence, consciousness and bliss).
When the rishis come down from this elevated spiritual state, they try to express their experience or vision in human language. So the vision is divine, and the words are theirs. But even the words are so unique and inimitable. And the rishi’s personality is also unique.
All this is seen in the Rig Veda itself. Also, the divine vision and expression of it in human language are at different levels of this Speech. This is seen in RV 1.164.45:
चत्वारि वाक्परिमिता पदानि तानि विदुर्ब्राह्मणा ये मनीषिणः ।
त्रीणि गुहा निहिता नेङ्गयन्ति तुरीयं वाचो मनुष्या वदन्ति ॥
catvāri vākparimitā padāni tāni vidurbrāhmaṇā ye manīṣiṇaḥ ।
trīṇi guhā nihitā neṅgayanti turīyaṃ vāco manuṣyā vadanti ॥
Four are the levels of Speech which the wise people know of. Three of them are hidden and do not stir, the fourth is what humans speak.
The Vedas are unique because they express the ultimate eternal truth that anyone can experience. It is the ultimate eternal truth that is apaurusheya. It is through expressing this apaurusheya truth that is available for everyone, that the Vedas are apaurusheya.
This is what Bṛhadāraṇyaka Upaniṣad (1.4.10):
...अहं ब्रह्मास्मीति । ... तद्यो यो देवानां प्रत्यबुध्यत स एव तदभवत्तथर्षीणां तथा मनुष्याणाम् ...। तदिदमप्येतर्हि य एवं वेदाहं ब्रह्मास्मीति स इदं सर्वं भवति ... ।
"I am Brahman"... whosoever among the Devas realized this, he became it. Likewise among the rishis and likewise among the common people. Hence, even today whoever realizes, "I am Brahman", he becomes all this...
Why is this eternal truth apaurusheya? It is independent of human relativity. It does not depend on a particular "prophet" who is the only one who can access this.
The Vedas are apaurusheya because they, uniquely among all other scriptures, express this apaurusheya truth.
The Quran fails under all these above criteria. The Quran claims that only Muhammad is God's prophet, and no other human being can become a prophet or connect with God. Claiming prophethood is blasphemy in Islam. So Quran's claim of authenticity is strictly dependent on the authenticity of Muhammad. So Quran is strictly paurusheya, i.e. strictly human-dependent. So if there is any fault or imperfection in Muhammad's character, then Quran's authority fails completely. (Which is incidentally why they are so defensive about Muhammad.)
However, the authority of Vedas is infallible because the eternal truth they express is not dependent on any specific human being. Anyone today can verify and confirm that the rishis expressed the eternal truth thousands of years ago. Hence, the apaurusheyatva of Vedas is upheld.
This is the only perspective from which you can "refute" the infallibility of Quran and uphold the infallibility of Vedas.