Does putting a counter on a creature with modular when it dies count as a triggered ability?

Modular is a keyword ability with two functions: it causes the permanent to enter the battlefield with a certain number of +1/+1 counters, and it includes a triggered ability that fires when the creature with modular dies. The triggered ability reads (in general): 'When this creature dies, you may put its +1/+1 counters on target artifact creature.' This is explicitly a triggered ability, not a static or activated ability (CR 702.43).

Because it is a triggered ability, it uses the stack (CR 603.2). When the modular creature dies, the ability triggers and is placed on the stack. Players receive priority and may respond before the counters are moved. The controller of the ability then chooses a target artifact creature and, upon resolution, moves the +1/+1 counters to it.

The 'enters the battlefield with counters' part of modular is a static ability that creates a replacement effect (CR 614.1c), so that portion does NOT use the stack. However, the death trigger is entirely separate from that.

Concrete example: You control Arcbound Ravager (modular 1) with three +1/+1 counters. It dies. Its modular triggered ability goes on the stack. Your opponent cannot prevent this trigger from occurring, but they can respond to it — for instance, by destroying your target artifact creature before the counters move. If the target is gone when the ability resolves, the ability is countered and no counters are moved (CR 608.2b).

In summary, yes — the counter-transfer from modular upon death is a triggered ability (CR 702.43, 603.1) that uses the stack and can be responded to.