Yes, its mentioned that Krishna was born in Rohini nakshatra while Rama was born in Punarvasu nakshatra
Srimada Bhagavatam Canto 10:Chapter 3
S'rî S'uka said: 'Then there was the supremely auspicious hour that,
with the constellation of Rohinî rising, was endowed with all
qualities. All the stars and planets stood in a favorable position.
Krishna's horoscope based on Surdas poem
Samvat saras vibhavan, Bhado aath tithi budhvaar, Krishan paksh Rohini, Ardh nisi harshan jogudaar, Vrish hai lagan, ucha ke nisipati, tanhi bahutsukh pe hai,
Chauthe sinh rasi ke dinkar, jiti sakal mahi lehey, Panchvey budh kanya ko jo hai, putrani bahut bade hai, Chatvey shukra tula ke shani jut datru rahan nahipe hai, Unch neech yuvati bahu kari hai, satvey rahu pare hai, Bhagye bhavan mei maker mahi sut, nahu aishvarya bade hai, Laabh bhavan mei meen brihaspati, nav nidhi ghar mei aihey, Karam bhavan ke iis, sanichar syam baran tan aihey, Aadi sanavan parbra ham prabhu, ghat-ghat antarjami, Sau tumhare avtarey aani ke, Surdass ke swami.
Lord Krishna's horoscope
Moon sign in Taurus of Venus planet made Krishna beautiful and loved by many like Gopis, Neechbhanga Ketu in ascendant made Krishna to liberate several souls in his lifetime. Sun sign in Leo and 4th house of wealth made Krishna to be powerful wealthy king.
Birth rashi of Lord Rama is mentioned in Bala Kanda of Ramayana as Punarvasu as Rashi is decided by Moon Sign in a horoscope.
Bala Kanda, Ramayana by Valmiki, Chapter (Sarga) 18
18:8,9,10,11 On completion of the ritual six seasons have passed by
and then in the twelfth month, on the ninth day of chaitra month
[April-May,] when the presiding deity of ruling star of the day is
Aditi, where the ruling star of day is punarvasu, the asterism is in
the ascendant, and when five of the nine planets viz., Sun, Mars,
Jupiter, Saturn, and Venus are at their highest position, when Jupiter
with Moon is ascendant in Cancer, and when day is advancing, then
Queen Kausalya gave birth to a son with all the divine attributes like
lotus-red eyes, lengthy arms, roseate lips, voice like drumbeat, and
who took birth to delight the Ikshwaku dynasty, who is adored by all
the worlds, and who is the greatly blessed epitome of Vishnu, namely
Lord Rama's horoscope
Moon sign in Cancer of dynamic Moon planet made Rama to do lot of travels in his life with Jupiter making him wise and forming Gajkesari Yoga with Moon, and Sun sign in Aries of Mars planet and house of profession made him natural king.
18:15 The fair-minded Bharata is born under Pisces where puSyami is the star
of day, and the sons of Sumitra, namely Lakshmana and Shatrughna are
born under Cancer, where ashlesha is the star of the day and when sun
Also, birth constellation of Bharata is Pushya with ascendant Pisces,and of twins Laxmana-Shatrughana, brothers of Rama is Ashlesha with ascendant Cancer. The lord and symbol of Ashlesha nakshatra is Naga or Hydra, which is also an another name of Sheshanaag, whose avatar were Laxman-Shatrughana. All three Pushyami, Punarvasu-4 and Ashlesha nakshatra are part of Cancer(Kataka) ruled by Moon and as the Moon sign of the 4 brothers Rama, Bharata, Laxmana, Shatrughana, hence made them highly compatible with each other.
For the full effect of constellation to take place luminaries(Sun and Moon) should be strong in degrees and in the Kendra(1,4,7,10) houses without combustion. For example in both the horoscopes of Rama and Krishna, Sun and Moon are at 90 degrees which shows that Rama was born on Shukla Navami(9th)(Moon is ahead of Sun by 4 houses) while Krishna was born in Krishna Ashtmi(8th)(Moon is behind Sun by 4 houses) which are around in the middle of Shukla and Krishna Paksha of 15 days. Full moon(Sun and Moon aspecting) or No moon(Sun and Moon in conjunction) causes combustion of Moon(mind) with Sun(soul) and is not regarded as auspicious as the middle of Pakshas. In these cases, ascendant is same as Moon sign, but usually for normal persons, Sun sign, Moon sign and Ascendant(Body nature) all three should be analyzed to predict about the attributes of any person.