IRL, medieval times, monk monasteries and clergy were often part of the same organization.
It's important to note however that the D&D monk is NOT a european type monk. It's more akin to the Shaolin and buddhist monks (chinese), which were very recluse.
A D&D monk monastery and a D&D cleric temple would have little to no reasons to associate because of their class, unless they worship the same god, which is another thing altogether.