I quit my old job in October and left the US in December 2018. I started a new job abroad in January 2019. In which tax year can I claim my moving expenses ?

Unfortunately, the moving expense deduction is no longer available in tax years 2018 – 2025 due to the Tax Cuts and Jobs Act (TCJA) that Congress signed into law on December 22, 2017.

TurboTax' Help is confusing: "If you’re an active-duty service member who relocated due to a permanent change of station (PCS), your move automatically qualifies. If not, your move will need to meet certain requirements.
To qualify for this deduction as a full-time employee, you'll need to work at the new location for at least 39 weeks in the 12-month period, once you've arrived in your new city. …."

If you are not active duty military, you can, unfortunately, no longer claim moving expenses due to the changes in the tax law.