Okay, your original example had month and year numbers in separate cells. If you want to reference a single column which contains dates then a different formula is required. Here's one variation:
Code:
=AVERAGEIFS(SUPPORT!K9:K258,
SUPPORT!J9:J258,">="&EOMONTH(TODAY(),-1)+1,
SUPPORT!J9:J258,"<"&EOMONTH(TODAY(),0)+1)