Can You Ask Only Happy Customers for Google Reviews?

No, and it is called review gating. Deliberately routing only happy customers to Google while steering unhappy ones elsewhere is against Google's policy and, in some places, consumer-protection law.

Ask everyone who genuinely used your service. A profile built honestly is more convincing anyway: a wall of nothing but fives with no substance reads as suspicious, while a strong average with the odd critical review reads as real.