Based on when the last child was born in the Continuum, we know that he's at least a millennia old. How old is he?
Following the death of Quinn, a massive Q Civil War broke out, as the forces of the status quo resisted the calls for change in the Continuum, by a faction led by Q himself. Seeking to end the conflict, Q devised a plan to mate with Kathryn Janeway, the captain of Voyager, in order to create a new Q/Human hybrid – a new breed of Q that would help bring an end to the civil war. Janeway, however, flatly refused.
Q then proposed mating with his old girlfriend (instead of with Janeway) and she agreed. The new child, nicknamed Junior, became the first child born in the Continuum for millennia, and his presence brought an end to the civil war. (VOY: "The Q and the Grey")
We know from this that 'the Q' can mate and do reproduce, so Q was in fact 'born'. He jumps back and forth along the timeline from the beginning of life on earth to the far distant future so that's no real help to pin down an actual age. Is he really just a 'child' by continuum standards, à la Trelane (TOS: "The Squire of Gothos") or as old as the ages?
How old is he really?
Some try to claim that all Q suddenly came into existence at the same instant 'forever ago', therefore this question is held as redundant and without merit. The 'forever before and after' Q position is generally rationalized because the Q are immortal, exceedingly powerful and can time travel. Just because a being can have breakfast in 2375 on a starship, hangout on earth for lunch watching people trying to learn flint-knapping in 15,000 BC (after misplacing an asteroid belt), then jump forward to 15,000 AD for dinner while watching people trying to learn flint-knapping doesn't mean they have always been.
Such a position is not only tenuously founded but ignores the contrary facts and evidence that Q have been born in the past and born in 'the present'.
Per Memory Alpha: "Based on descriptions by numerous Q, including the Q later known as Quinn, as well as Amanda Rogers (the child of two Q in Human form), the Q Continuum is a highly ordered society, but is also the result of eons of evolutionary stagnation." http://en.memory-alpha.org/wiki/Q_Continuum
This alone admits that the Q are in fact the result of evolution. Because of this current state of 'evolutionary stagnation', Q even admits to Riker during his temptation (TNG 1X10 "Hide and Q"), that humans could become the Q's equal in the distant future. Also because of this 'evolutionary stagnation', Junior's conception caused quite a stir. It hadn't been done in quite some time...
All this side issue and Continuum lore being interesting an all still doesn't address the base question posed; How old is Q (John de Lancie)? Has there ever been any hint or comments relating to HIS age?
de Lancie himself discusses that the 'possible' creative force behind the 'Q' character was very likely the 'Trelane' character from TOS; who by the end of the episode was found to actually be a child from an ultra advanced/evolved species. Many believe that Trelane was in fact a 'Q' though he needed to use a hidden power source for much of his antics.