If one says (with no intention to make a vow), "I shall fast for one day with the purpose of making Allah happy", would his saying be considered a vow?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

A vow is not effective except if it is said in a form that makes one feel bound by it, such as saying: “It is due upon me to Allah to fast or give in charity such-and-such amount”, or “If such-and-such happens, then I must fast such and such day”, for example.

Ibn Qudaamah said in al-Mughni:

“The wording of a vow is to say, ‘It is due upon me to Allah to do such-and such.’ If the person says, ‘I vow to do such-and-such’, then the vow is also binding because he explicitly mentioned it. It is also considered a vow when he says, ‘If Allah cures me, I will fast a month.’” [End of quote]

Based on the above, the saying that you mentioned in the question is not a vow.

Allah Knows best.