First to say, Quartz crystals are very reliable when compared with other components. They almost never brake. The only time I've seen a defective one was in a drone that felt from a height of about 20-30 meters, and the crystal shattered at impact.
Most probably the Quartz will be the last thing to check on a motherboard. The most common failure for a motherboard is bulged or high ESR electrolytic capacitors.
Anyway, if a quartz is bad, it usually won't oscillate at all, so if you see other digital signals changing, then the quartz is good. There is no such problems as a quartz is oscillating, but the amplitude is not enough.
However, if you still want to check a quartz oscillator, the best way is to measure this indirectly on another digital output, not by directly probing the quartz. If you don't have such a digital output available, or you insist to probe the quartz directly, you can use an active probe. Alternatively, to see if it oscillate without disturbing it much, try a small wire loop on the tip of the probe, acting like an antenna, and put it near the oscillator without touching the circuit. Another technique could be to measure through a very small capacitor in series with the tip of the probe, about 1pF or less.
In any case, the amplitude is irrelevant, and it is not something to worry about. If you have, let's say a 20MHz quartz and you see some oscillations at the expected 20MHz frequency, no matter the amplitude, the quartz is good.