Propane Hose Adapter Question

That is a propane quick connect fitting. This is NOT the same as a compressed air quick connect fitting. Be alert that, into the bargain, TT makers have used two different incompatible quick connect systems, which, along with the compressed air one all look identical. They are just enough different that you cannot connect one to either of the others.

I have done exactly what you want to do, i.e., connect a standard BBQ grill to that connector, and to do it, you will have to remove the regulator on the grill, as the propane line connected to that fitting is a low pressure line. You need the appropriate propane hose, the right QC (male) connector, and such plumbing as needed to tie into the grill.

You will need to get a hose fitted with the quick disconnect portion shown on right side of attachment. Should be available at most propane retailers.
Quick disconnect
Note that the grill must be compatible with pressure at disconnect. Grills require high pressure.

If that propane hose is from the trailer regulator which it probably is then you cannot hook it to a grill that has a regulator on it. You will need to either bypass the trailer regulator or get a grill that does not have a regulator on it.