Using the cyclones internal clock as input for a circuit

I'm not sure how to respond since it's not really clear to me what you are trying to accomplish. A typical counter has a clock input which all synchronous operations (i.e. counting) operate on. You prevent the counter from incrementing by using a counter enable which enables the counter register to capture the new value which is the output + 1. If your existing logic doesn't have these I highly recommend structuring your counter around that since it'll make your life much easier. It might be worth looking at the "lpm_counter" megawizard function since it's structured around what a typical counter provides in terms of behavior and once you use something like that then performing time measurements becomes trivial.

From your desription it almost sounds like you were using the push button as the clock source previously. You do not want to do that in a FPGA design for various reasons, instead you should be using an actual periodic clock and register enables to enable/inhibit synchronous operations from occuring. I'm sure if you search around you'll find plenty of schematics of counters since I'm guessing plenty of schools have students doing this sort of thing.