The residence law says that one of the requirements for getting a permanent residence permit is being a holder of residence permit for some period.
For the Blue Card, the same paragraph makes an exception and says that the Blue Card holder needs to have worked and payed taxes for a period of 21-33 months, see §18c AufenthG (2) for details.
In Berlin, they say that they can count the time when your job satisfied the requirements of the Blue Card, so you have good chances to get your months counted from the start of the job contract. Other lands, however, might have different understanding of this law.
IMHO, the difference of 2-3 months is not that important. First, one can try to apply for permanent residence permit a couple of months earlier, since the processing usually takes at least 1-2 months. Second, Auslaenderbehoerde might still have reduced opening hours or do not accept any applications at all due to COVID-19. In the end, it all depends from the particular person there, if she will accept an application right now or ask to come a couple of months later.
P.S. It is also worth mentioning (but not directly related to your question) that generally there are many other requirements for getting permanent residence permit, such as knowledge of German, paying taxes for some period (taxes payed by spouse might be counted as well), etc.