There is a tradition - or whatever you may call it - not to tell people about your pregnancy early on.
First off, is there a Torah basis to it? Or just common sense? (if things go wrong people will still say Mazal Tov...)
Secondly, when someone asks "are you pregnant?" or "is your wife pregnant?", is OK to lie and say NO??
Here is an excerpt from Kovetz Minhagim, An Anthology of Chabad Lubavitch customs regarding pregnancy, childbirth, circumcision, redemption of the firstborn, and the birth of girl:
It is the custom of chassidim who are careful to conduct their lifestyles according to the practices of old that they conceal their wives' pregnancies until they have entered the fifth month. Undoubtedly, this practice has an inner foundation. Of course, the precaution is about publicizing the pregnancy. It does not apply to divulging it to very close relatives without broadcasting it.
See there for sources in the footnote. Of interest is footnote 3, which quotes a letter of the Previous Lubavitcher Rebbe:
"The accepted practice is that until three months of pregnancy it is concealed even from very close relatives..."
As a partial answer, concerning the lying part, it may be adduced that it is permitted to lie from Rema (resp. §2) who permitted a woman to have contact with her husband despite her telling inquirers, early on in her pregnancy, in order to be shielded from the evil eye that she wore clothing designated for her menstruation period because she was unclean and indeed said she was unclean. He does not mention the (non)issue of her lying. This ruling was also brought by Shach (YD 185:5) and likewise did not raise the issue of lying.
R. Shimon Hirari (resp. Lev Simchah, EH §12) justified a woman's white lie, which she told people that she had a ruach (I can't determine if she meant a spirit or bloated), by demonstrating that pregnancy can be referred to as ruach.
However, in both cases it should be noted the woman did not explicitly lie by saying she is not pregnant but rather gave "alternative facts".
This custom is much wider than the Jewish world.
Most women do not widely publicize their pregnancies until the third month because the rate of, G-d-forbid, miscarriage drops significantly at week 12. By month 5, women start to show, so it becomes nearly impossible to keep it a secret.
However, my wife would ream me out my if I ever lied as a response to that question. (She would say that I should have changed the conversation topic.) She'd say that saying "no" is basically telling the ayin hara you don't want it.
