Generally speaking, yes.
If it is the start of a new fiscal year and you are within the cap of an H-1B visa, then you can apply for it. If your employer is exempt from the cap then you can apply anytime.
However, the tricky part comes if your J1 has the two-year residency requirement. If it does, then you must live in your home country for two years before applying for the H-1B. If you have this requirement, then you could apply for a waiver. If approved, the waiver would essentially revoke the need for you to move to live in your home country for two years.
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