No. Claiming changes who controls the account, not who has access to Jira, and the Jira bill only counts people with access.
Two separate things are going on here.
Your Jira subscription counts the users who have product access to Jira at the end of the billing cycle (Atlassian Administration → Billing → your Jira subscription shows the number). Whether those users are managed or external makes no difference to that count. Your team members are already in it today because they already use Jira. Claiming them adds nobody.
Claiming moves an account from "external" to "managed". You get to edit, deactivate or delete it and apply policies to it, and the person can no longer change their own email or delete their own account. Product access stays where it was. If you also claim accounts on your domain that don't use Jira at all (someone who signed up for Trello with their work email, say), they become managed too, but they have no Jira access, so they still aren't billed.
The one place a managed account has a price is Atlassian Guard Standard, and only if you subscribe to it. Even there Guard counts external users with app access the same as managed ones, so claiming people who already use Jira doesn't move that number either.
https://support.atlassian.com/subscriptions-and-billing/docs/manage-users-and-user-tiers/
https://support.atlassian.com/user-management/docs/verify-a-domain-to-manage-accounts/
https://support.atlassian.com/subscriptions-and-billing/docs/manage-your-bill-for-atlassian-guard-standard/
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