In 5e, the Tarrasque has an Intelligence score of 3, and its other two mental scores are nothing to write home about either. This has been consistent throughout the editions, implying that it is an animalistic creature that acts on instinct. However, the Tarrasque is no ordinary animal, it is a monstrosity, and it's instinct seems to be one of pure hunger and destruction. The 5e Monster Manual has this to say about the Tarrasque:
Its cavernous maw yawns wide enough to swallow all but the largest creatures, and so great is its hunger that it can devour the populations of whole towns. (...) When it awakens in answer to some inscrutable cosmic call, it rises from the depths to obliterate everything in its path.
So the Tarrasque's instinctual hunger is probably not entirely natural, but part of its magical nature. Not much is known about the Tarrasque or how it came to exist, but there are several official theories. The 4th edition Monster Manual suggests that the Tarrasque was created by the Primordials as a weapon of destruction during the Dawn War (a war led between the primordials and the gods) and that only one was created, the one that is now slumbering in the Prime Material Plane's core in the official (Forgotten Realms) setting. Since there is only one, it likely lacks the ability to reproduce sexually, and that would prevent it from understanding or feeling that sort of love. However, the official Spelljammer setting (which is also part of the Forgotten Realms canon) suggested in Practical Planetology that there is a planet called Falx inhabited by entire species of Tarrasques, who can presumably reproduce amongst themselves somehow. Since Spelljammer is quite a wacky setting and hasn't had any major content released for it in 5e, the Dawn War interpretation is probably the "more canon" one.
Now, that possibly rules out romantic or sexual love, but many other kinds of love and affection exist. The Tarrasque's low Intelligence score by itself shouldn't prevent it from understanding or feeling love, since dogs in D&D are represented by the equally as "dumb" Mastiff stat block and we all know how affectionate dogs can be. The primary concern is whether the Tarrasque's instinctively destructive nature programmed into its very being would get in the way of that, and I don't think there's a canon answer. Personally, I think that with a lot of patience and skill, a creature could get the Tarrasque to care about it enough not to eat it as long as it was offered other food regularly, like a trained wild beast, but if the food ever ran out or the Tarrasque somehow became angered, well...
It is worth mentioning that the D&D Tarrasque is based on an actual folkloric creature from a French legend: https://en.wikipedia.org/wiki/Tarasque#Legend . In the legend, the creature devastates the landscape far and wide, and it cannot be stopped by any military force the kingdom can muster. However, a gentle saint strong in her faith succeeds in pacifying it with prayers and hymns to the point where the creature doesn't even defend itself when the frightened folk attack and slay it. So teaching a Tarrasque love might actually be very true to the monster's origins. Maybe a little chin scratch performed with sincere affection is all the Tarrasque needs to overcome its desire to devour and obliterate everything in its path implanted in it by its hateful creators...
