Flops count of (sone of the) LAPACK subroutines is given in Appendix C of the Installation Guide for LAPACK, LAPACK Working Note 41, Susan Blackford and Jack Dongarra.
For DORMQR go and look at the bottom of page 12, you have two cases SIDE=R or SIDE=L.
And FYI, the cost of DORMQR is NOT the cost of DORGQR (generating Q) + the cost of DGEMM (multiplying with Q), it is significantly less.
In particular since you seems to indicate you only have one vector (since you say the multiplication is 2mn) to multiply with, forming the
whole Q would be a big time consumption and a big memory requirement.