Often, but not always
Many ability checks in combat require Actions, and many Actions require ability checks. Unambiguous cases are spelled out in the section on Actions in Combat (PHB, p. 192-193). For example, the "Hide" or "Search" actions both require ability checks, and both take up your Action (unless you have a special feature which allows you to do them some other way).
However, not every ability check in combat requires your full Action. A prime example is grappling. (PHB, p. 195, bold added)
When you want to grab a creature or wrestle with it, you can use the
Attack action to make a special melee attack, a grapple. If you’re
able to make multiple attacks with the Attack action, this attack
replaces one of them.
The target of your grapple must be no more than one size larger than
you, and it must be within your reach. Using at least one free hand,
you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target’s Strength (Athletics) or
Dexterity (Acrobatics) check (the target chooses the ability to
In this case, we have two ability checks which do not require a full Action. One on the part of the grappler, which requires the use of one of their attacks, and another on the part of the target of the grapple which requires no action, reaction, or movement. (Of course, the grappler had to use the Attack Action to be able to grapple at all, but the ability check to grapple did not use up the entire Action).
Another example is found in the spell Tsunami (PHB, p. 284, bold added):
A creature caught in the wall can move by swimming. Because of the force of the wave, though, the creature must make a successful Strength (Athletics) check against your spell save DC in order to move at all. If it fails the check, it can’t move. A creature that moves out of the area falls to the ground.
If this check required an action, the spell's description would say so. Contrast this with the description found in the spell Wrathful Smite (PHB, p. 289, bold added):
As an action, the creature can make a Wisdom check against your spell save DC to steel its resolve and end this spell.
Does it take an Action? DM decides
As a general guideline, it's heavily implied that more difficult activities require actions: this is especially stressed in "Other Activity On Your Turn" (PHB, p. 190, bold added):
The DM might require you to use an action for ... activities when it needs special care or when it presents an unusual obstacle. For instance, the DM could reasonably expect you to use an action to open a stuck door or turn a crank to lower a drawbridge.
And it's understandable that you would think that all activities that require ability checks require an Action. After all, the definition of ability checks states (PHB, p. 186, bold added)
The DM calls for an ability check when a character or monster attempts
an action (other than an attack) that has a chance of failure.
But this definition states "when you use an action that could fail, it requires an ability check": the converse, "when you use an ability check, it requires an action," is not necessarily true. If you attempt something in combat which is defined as taking an action and requiring an ability check, then it definitely requires both. Otherwise, the sections on both "Other Activity On Your Turn" (PHB p. 190) and "Improvising an Action" (PHB p. 193) leave the decision up to the DM when to call for the use of an action, the use of an ability check, both, or neither.