Currently ".nyc" TLD is not in the list of Domains that you can register with Amazon Route 53. AWS does keep adding new TLDs to the list so keep an eye on the list of officially supported TLDs.
That being said, as the domain is already registered with another registrar, you can just:
Create a public hosted zone in Route53: https://docs.aws.amazon.com/Route53/latest/DeveloperGuide/CreatingHostedZone.html
Get the nameservers for this zone: https://docs.aws.amazon.com/Route53/latest/DeveloperGuide/GetInfoAboutHostedZone.html
Set these nameservers in your registrar console (often found in glue records or nameservers)
Use your public hosted zone by creating records
Note here that you can have a registrar (for example GoDaddy) different than the DNS service (Amazon Route53).
See this link that explains the whole procedure: https://docs.aws.amazon.com/Route53/latest/DeveloperGuide/migrate-dns-domain-in-use.html
Hope this helps.
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