Is it permissible for me not to observe hijab before the husband of my husband's daughter or not?.
The husband of your husband's daughter is not a mahram for you, because there is no reason to suggest that he is a mahram. Allaah says (interpretation of the meaning):
“Forbidden to you (for marriage) are: your mothers, your daughters, your sisters, your father’s sisters, your mother’s sisters, your brother’s daughters, your sister’s daughters, your foster mothers who gave you suck, your foster milk suckling sisters, your wives’ mothers, your stepdaughters under your guardianship, born of your wives to whom you have gone in — but there is no sin on you if you have not gone in them (to marry their daughters), — the wives of your sons who (spring) from your own loins, and two sisters in wedlock at the same time, except for what has already passed; verily, Allaah is Oft‑Forgiving, Most Merciful.
Also (forbidden are) women already married, except those (slaves) whom your right hands possess. Thus has Allaah ordained for you. All others are lawful”
[al-Nisa’ 4:23, 24]
Based on that, you have to observe hijab before him, and it is not permissible to you to uncover in front of him or to be alone with him, because he is a stranger or non-mahram like all other non-mahrams.
As for the husband of your daughter, he is a mahram for you, because Allaah says (interpretation of the meaning): “your wives’ mothers”.
And Allaah knows best.