I would try using Dates (from the standard library docs here: Dates · The Julia Language)
For your example:
using Dates
dr = Date(2000,1,17):Day(1):Date(2020,1,17)
count(x -> monthday(x) == (2, 29), dr)
Also there’s:
isleapyear(2020)
This should address all of the complications of leap years (2020 is a leap year, 2100 is not, 2000 is)