Yes, Chataka and Chakora birds are mentioned in our Scriptures like Itihasas and Puranas. Some example are given below:
Devi Bhagavat Purana, Book 9, Chapter 2
Her eyes, also, like the Chakora bird (Greek partridge) began to drink incessantly with joy the moon beams from the face of S’rî Krisna, defying, as it were, the ten millions of moons.
Mahabharata: Book 3: Vana Parva: Section CLXXVII
And then he beheld (some) lovely spots in the Himalayan mountains, frequented by Devarshis and Siddhas and inhabited by hosts of Apsaras, resounded here and there with (the warbling of) birds--the chakora, the chakrabaka, the jibajibaka and the cuckoo and the Bhringaraja, and abounding with shady trees, soft with the touch of snow and pleasing to the eye and mind, and bearing perennial fruits and flowers.
Mahabharata: Book 12: Santi Parva: SECTION CCCLX
As the bird called Chataka, which is fond of rain, waits in earnest expectation of a shower (for slaking its thirst), even so is that Brahmana waiting in expectation of a meeting with thee.