Hi there, and congratulations on your new budgie boy!
From that photo it's clear your budgie is much older than 1 month old, he has completed his first moult, already has all of his big boy throat spots and has the mature royal blue colour on the cere. Taking all these factors into account, your budgie is at the very least 6 months old, there is no doubt on my mind.
What I don't get is why people lie about ages? Or are they just not good at knowing? It took me a few months but I can definitely tell the rough age of a budgie now. People make me wonder daily as to why they do certain things