Short answer: Usually not. COBRA does not apply to disability coverage, and group STD and LTD normally end with employment. Some LTD contracts offer a portability or conversion option, but that depends entirely on the policy.
COBRA reaches only group health plans, defined as plans providing medical care (29 U.S.C. 1167(1)). Disability income is not medical care, so there is no COBRA right to continue STD or LTD.
Some group LTD contracts include a portability provision that lets a departing employee who is not disabled continue coverage, often at a different benefit and rate, by applying within a short window after coverage ends. Others offer conversion to an individual policy. Many offer neither, and STD rarely does.
A claim that began while the person was covered is different: LTD benefits for a disability that started during employment normally continue under the policy after employment ends. For employees who want income protection that travels with them, an individually owned disability policy is the usual answer.