No, significant figures of a physical quantity do not change if we measure a physical quantity in a different system of units.
E.g. 3.23 inch = 8.20 cm
In both cases we see the number of significant figures is same.
A confusion may arise here
1 foot = 0.3048 meter exactly
Apparently this may mean for one significant figure for foot it resulted in 4 significant figures in meters. But this representation is erroneous because 1 meter is accurate .
Now if 1 m is written as 3 significant figures in this way
1.00 m = 0.30 ft.