Don't make it easy on yourself by requesting me to read the original request again.

The actual quote was:
"this loop should work in background(polling function)that is microcontroller should count input pulses as well as cheak one minute is over or not."

No mention of the processor doing anything else but waiting and counting input pulses. In short, a traditional timeout delay. Remember that a delay loop need not consist of just dummy instructions and a test if the time is up. It can test a large number of things - such as polling a device for availability of input data or counting pulses.

And while possible - but unreliable - to do with pure C, it is also most definitely possible to do with other timing means than a timer.

"Impossible" is still a very strong word to use. Remember my mention of a synchronous transfer with regular transfer interrupts? That manages fully background timing without an explicit timer. A USB device can produce a 1ms interrupt - or count a 1ms frame counter - while still not being a timer. If you run out of timers, there may a number of peripherials that can be put in loopback mode to either produce a known frequency or create and/or drive a counter.

In the end, you may have an existing product that may either be (ab)used to solve the requested job by a bit of creativity. Or you may say "impossible" and request the processor to be replaced, resulting in a large delay in shipping and possibly new certification tests. So the "impossible" should be saved until you have spent some time making sure that you really can't see other alternatives!

In this case, the OP did not supply enough info for you to:
1) Know what other work needs to be done while performing the 60s wait.
2) Know what other hardware devices or external stimuli that is available.