Hello,
You can look at https://docs.atlassian.com/atlassian-bamboo/REST/6.8.1/ and you need to use resource /queue.
Sample will be like:
curl -u admin:admin -X POST http://BAMBOO_URL/rest/api/latest/queue/TEST-TEST
For deployment use resource /queue/deployment. But at first you need to create release. It's also possible by REST API
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