Elgg does not use SMTP to send email notifications. Only if you use some additional plugin it might be that your Elgg site sends the mails via SMTP. In this case the SMTP settings would have to be changed in the plugin settings of this plugin.

What exactly do you mean by the transfer to a new server location? Has only the server changed with the domain name being the same? What is used as site email of the site (see Basic Settings)? If the domain name has changed and the site email address is from the old domain the server might deny sending the emails. If the domain name has not changed (and site url etc. also hasn't to be changed) the question is if the email account used as site email exists on the new server and if the MTA (mail transfer agent, e.g. Postfix) is set up on the server. The MTA must be set up for sending of mails to work as Elgg itself relies on it and has no email server capabilites itself.