You can perform Blast of your primers using NCBI Primer blast tool...

http://blast.ncbi.nlm.nih.gov/Blast.cgi?PROGRAM=blastn&PAGE_TYPE=BlastSearch&LINK_LOC=blasthome

Normally for qPCR you'd target a specific gene for which you've designed primers to amplify a small fragment (100-300bp), spanning at least one exon-exon boundary.

You can BLAST your primer sequences, but really you should know where they'll amplify anyway if you've designed them correctly.

So I assume you're trying to find out whether your primers might amplify non-specifically by binding to and allowing amplification of a non-target mRNA.  I'm not sure if there is a specific tool for this - you could BLAT at ncbi against ESTs or mRNA sequences but you'd have to do a lot of manual annotation if you had a lot of potential primer pairs.  Regardless of the outcome of such a tool, you should run a normal thermocycler PCR under your qPCR cycle conditions before you begin your qPCR assay and check for non-specific bands on an agarose gel (only visible bands should be significant).  You can also purify this product and dilute to use as your top standard for qPCR.

My approach would be to choose 4 different pairs and run a regular PCR under qPCR conditions for each pair and check for specificity, then run the standard curve for those which were specific and choose the pair with the best curve.

Hope this helps and isn't too patronising :S

I have found these  links for designing or may be checking primers for miRNA.

Experts in the same field and suggest if these tools are reliable.

Following are the links.

http://genomics.dote.hu:8080/mirnadesigntool/

http://www.leonxie.com/miRNAprimerDesigner.php#

https://www.exiqon.com/mirna-qpcr-designer