I'm not sure if you intend to replace the axles themselves, but AFAIK, most nutted axles are not hollow, which means they can't accommodate QR skewers. If this is true in your case, you will need to replace the axles outright in order to use QR skewers.
QR axles for modern hubs do come in a few "standard" sizes, but you will nevertheless want to measure the dimensions of your frame and hubs anyway, in case they are not standard. You mentioned your bike is second-hand, which indicates it might be older, in which case this is doubly so.
There are three dimensions to check:
Hub spacing - This is the distance from the inside of one dropout to the other. You can measure this by removing the wheels from the frame and sticking a ruler in the dropouts. To replace the axles, you will need to find QR axles and skewers to match your hub spacing.
Otherwise, the QR may be too short or too long for the hub. With non-QR axles, an axle that is too long is not a problem (the nuts will still go on fine, and if you need to, you can cut the excess axle), but with a QR axle, you can't really trim the axle, because the QR skewer can't be trimmed.
Typically, hub spacing on modern bikes is 100 mm (front) and between 120 and 130 mm (rear). Again, you should measure this yourself-- I'm just giving you an idea of what these numbers should be.
Axle diameter - The QR axles you get should match the diameter of the existing axles so you can re-use the existing cups and locknuts (unless you were planning on replacing those, too).
Common axle diameters are between 8 to 10 mm, or 5/16", or 3/8".
Thread pitch - This is basically the angle of the threads on the axle. Like the diameter, this will need to match the existing axle if you want to re-use the cups. According to The Sheldon, thread pitch is measured in either threads per inch or millimeters per thread, with a 1 mm/thread pitch being equivalent to 25.4 TPI.
Common thread pitches include 1 mm/thread, 26 TPI, and (rarely) 24 TPI.