The 1962 Chevy 235 would have had a timing gear (fiber) driven by a smaller crankshaft gear. Knowing the design as I do, it is nearly inconceivable to me either of those gears could have jumped a tooth or two. At least I'll say in 60 years I've never seen this happen, although I have seen a couple of fiber gears break and when that happens often a couple of even more serious problems follow.
If there is any way to remove the radiator and gain enough access to the front of the engine, you could remove the harmonic balancer (you'll need a puller), remove the front cover (you'll need also to remove the oil pan as it is likely you'll have 2 bolts/nuts that can't simply be unscrewed from the front...the two at the bottom) and rotate the crank by hand until you can see if the two dots...one on the cam gear and one on the crank gear...line up. But I wouldn't do this yet. A backfire is simply a spark plug igniting fuel at the wrong time...usually in the intake manifold or the exhaust manifold (which actually should be called something like after-fire but many folks still call it backfiring)...that is to say unburnt fuel is being ignited at the wrong time. Could be a valve stuck open or not working correctly. Could be ignition timing but if it is that far off, often the engine won't start at all.
Remove the valve cover and start the engine. See if all the rocker arms are working and if the valves are also opening and closing. Check your manifold vacuum with a gauge. If you have a valve sticking open or not closing, you'll see it clearly. The needle will drop every time the event occurs...vacuum fluctuations of a couple of inches could indicate ignition or ignition timing trouble and fluctuations of 4 or 5 inches could indicate a sticking valve. Let's start there and see where we go.
