There is no prohibition involved in others realizing someone is immersing, and it is permissible to tell select people as necessary in order to facilitate immersion.
Maintaining privacy is still important emotionally, though, so it is permissible to use white lies as necessary if that will help preserve your sense of privacy. This may depend on the specific shelter-at-home rules in your community and individual family dynamics.
Since the trip to mikveh is typically much shorter now, with all preparations done in advance, this might go more smoothly than you think. Saying you are going out to take a walk might suffice.
Also, keep in mind that you can prepare any time on the day of immersion, and break up your preparations at your convenience. Both of these steps can help preserve privacy. If you prepare earlier in the day, shower quickly and comb your hair right before leaving for the mikveh.
If you wish to talk out your particular case, please call our telephone hotline or your local Yoetzet Halacha.