Are there any early records between 1700 and 1760 of birthdays? And, where do I find them? I am particularly interested in marriages that happened in Belfast.
You are more likely to get baptism dates rather than birth dates, for the 1700s. There are some records. You need to say what precise denomination you are interested in. This link explains what records exist for Belfast:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
Most of the surviving records have been copied and most are in PRONI. A personal visit is required to view them.