Connecticut issues no painting licence. As with roofing, painting an existing home is home improvement work and is covered by the contractor's Home Improvement Contractor registration rather than by a trade licence of its own.
Try a free lookup — no signup, right here on this page. For programmatic access, take a free API key — 250 requests a month, no card. Also on RapidAPI and, if you would rather not write code at all, the Apify Store.
Ask for the HIC number. Connecticut requires it for home improvement work regardless of the trade, and it verifies here in one call.
Full coverage detail, including the caveats that apply to every Connecticut lookup, is on the Connecticut page.
The API is on RapidAPI with a free tier: Contractor License Verification. It is also on the Apify Store: Contractor License Lookup & Verification. Full schema at /docs.