No, that is NOT true.
While you are on OPT, you are under the F-1 visa and are considered to be a student. Hence, you are
exempt from FICA (Social Security and Medicare) taxes.
The day your H-1 visa becomes effective is the day you start paying FICA (Social Security and Medicare) taxes.
Download IRS Pub 519 from
Internal Revenue Service, print out page 51 and go to your employer's payroll to politely ask for a refund of the FICA taxes withheld to date and to stop further withholding until your H-1 visa becomes effective.