No, you cannot. There are no vowels.
Girlie
Those letters can be used to spell "quarreled".
Q; how do you spell trafficed A;trafficked
The correct spelling of the verb and adjective is equal(the same, equivalent).
quit
Quality, q-u-a-l-i-t-y.
http://letmegooglethatforyou.com/?q=m%5E3+in+L
avail
A seven letter word using the letters would be: virtual.
IQ
There is no seven letter word that can be made from those letters. Most English words that are spelled with a 'q' require a 'u' to follow it.
The word "queue" (pronounced "Q").