Although it’s often said to be from the Buddha, “It is possible to live happily in the present moment. It is the only moment we have,” is a quote from Thich Nhat Hanh. The first part in particular — “It is possible to live happily in the present moment” — is often found in his books.
The Buddha actually had very little to say — as far as we can tell from what’s been recorded in the scriptures — about the present moment. I don’t think he said anything about the present moment being all we have, although he might well have agreed with the statement.
In “No Death, No Fear” Thich Nhat Hanh writes:
The Buddha said, “It is possible to live happily in the present moment. It is the only moment we have.”
In another book he equates “It is possible to live happily in the present moment” with the Pali expression, Diṭṭha-dhamma-sukha-vihārā. Thanissaro translates this as “abiding in ease, here and now” and Bhikkhu Bodhi similarly renders it as “a pleasant abiding here and now.”
Diṭṭhadhamma means “this world” (literally “visible things”) and sukhavihārati is “dwelling happily (or at ease).” There is a convention that diṭṭhadhamma refers to time as well as place, hence “here and now” rather than just “here.” I don’t know why this is, but if both Bhikkhu Bodhi and Thanissaro include “now” as well as “here” then I presume they have good reasons for doing so.
However you translate diṭṭha-dhamma-sukha-vihārā, you’re never going to get close to “It is possible to live happily in the present moment. It is the only moment we have.” I suspect that what’s happened is that one of Thay’s talks was transcribed, and the editor, seeing “The Buddha said” inserted quotes around what he or she erroneously took to be direct speech.
As far as I’m aware, it’s a long time since Thich Hanh Hanh has written his own books, and the works published under his name are actually ghostwritten by disciples, based on material from his talks.
My guess is that Thich Nhat Hanh’s original intention would have been to say something like this:
The Buddha said it is possible to “live happily in the present moment.” It is the only moment we have. [Everything outside the quotes being Thay’s own words, of course, and the part within quotes being a direct quotation from the scriptures—a rendering of diṭṭha-dhamma-sukha-vihārā.]
If that’s the case, then this is the second instance I’ve found of a Fake Buddha Quote being created by Thich Nhat Hanh’s editors or ghostwriters, the other being “The moment you know how your suffering came to be, you are already on the path of release from it.”