Yes, in certain circumstances Qatar can prosecute you for crimes committed abroad. Article 17 specifically states that Qatari law applies to any person who resides in Qatar after committing certain serious offences abroad — such as drug trafficking or human trafficking — whether as a perpetrator or an accomplice. If you fall into this category, you could face prosecution upon taking up residence in Qatar.
More broadly, Article 18 provides that Qatari citizens who commit acts abroad that qualify as a felony or misdemeanor under Qatari law can be punished upon their return to Qatar. While this article refers specifically to citizens, expats should not assume they are entirely shielded, particularly for internationally recognised serious crimes.
The important protection available is the principle of 'non bis in idem' under Article 19, which means you cannot be tried again in Qatar for the same offence if you have already been finally acquitted, convicted, or have served your sentence in a foreign court. If you are concerned about your specific circumstances, consulting a qualified Qatari criminal lawyer before relocating or while residing in Qatar is strongly recommended.
This is general legal information, not legal advice. For advice on your specific situation, consult a lawyer licensed in Qatar.