All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and messenger.
The right of the parents is great as Allaah associated it with the command to worship Allaah Alone (Tawheed), as Allaah says (what means): {And your Lord has decreed that you not worship except Him, and to parents, good treatment...}[Quran 17:23]
Moreover, Abu Hurayrah narrated that a man came to the Prophet and asked him: “Who deserves my good company most?”, and he [the Prophet ] replied: “Your mother.” The man asked: “Then who?” The Prophet said: “Your mother.” The man asked again: “Then who?” The Prophet again replied: “Your mother.” The man asked again: “Then who?” Thereupon, the Prophet said: “Then your father.”” [Al-Bukhari and Muslim]
For more information, please refer to Fataawa 87019 and 82254.
A form of dutifulness to the mother is to spend on her if she is in need of that, but if she is not in need – as you mentioned - then it is not an obligation for the son to give her money, but it is a form of kindness to give her if this does not lead to causing trouble for him in terms of fulfilling his financial obligations like paying an immediate debt, or being neglectful in obligatory maintenance, like the financial maintenance of the wife and children.
The best way to be kind and dutiful to the mother is for the son to give her some of his money to please her according to his ability, as Allaah does not burden a soul beyond its ability. Nonetheless, he should apologize to her in a wise and gentle manner while always being kind and dutiful to her through various means.
However, it is not an obligation for him to inform his wife about what he gives to his mother, and he can resort to Tawriyah (i.e. saying something which has more than one meaning and intending a meaning different from what the listener is likely to understand) if she asks him about that in order to avoid disputes.
At any rate, the wife has no right to interfere in what her husband does in this regard as long as he fulfills his duty toward her.
Allaah Knows best.