Yes — private-sector employees can apply for most government recruitments if they meet the eligibility criteria.
You do not need permission from your employer to apply.
If you are selected, you will have to resign from the private job before joining.
Plan leave or time off for the exam, skill test and document verification dates.
Check the notification — some posts ask for an NOC at the DV stage.
Never skip reading the service rules and conditions of the post you apply for.