Short answer: No. Your monthly premium is what you pay to have coverage, and it is separate from your deductible and out-of-pocket maximum. Only your cost-sharing, like deductibles, copays, and coinsurance for covered care, counts toward those limits.
It is a common mix-up. The premium keeps your policy active; the deductible and out-of-pocket maximum track what you pay when you actually use care. The out-of-pocket limit does not include your monthly premiums, so paying premiums all year does not move you closer to either limit.
Federal rules draw the same line: cost-sharing includes deductibles, coinsurance, and copayments, but excludes premiums, balance billing amounts for non-network providers, and spending for non-covered services. That is why you can pay premiums every month and still owe your full deductible the first time you need care.