It's worth noting the comment table has a 'uid' column so you could potentially query that to get the comment count for a user.
e.g:
$user_comment_count = db_select('comment', 'c')
->condition('uid', $account->uid)
->countQuery()
->execute()
->fetchField();
However, if you want to increment a field on comment insert, you could use Rules or hook_comment_insert
e.g: Rules export (Increment field_comment_count for current logged in user on comment insert)
{ "rules_increment_comment_count" : {
"LABEL" : "increment_comment_count",
"PLUGIN" : "reaction rule",
"OWNER" : "rules",
"REQUIRES" : [ "rules", "comment" ],
"ON" : { "comment_insert" : [] },
"IF" : [
{ "entity_has_field" : { "entity" : [ "site:current-user" ], "field" : "field_comment_count" } }
],
"DO" : [
{ "data_calc" : {
"USING" : { "input_1" : [ "site:current-user:field-comment-count" ], "op" : "+", "input_2" : "1" },
"PROVIDE" : { "result" : { "result" : "Calculation result" } }
}
},
{ "data_set" : {
"data" : [ "site:current-user:field-comment-count" ],
"value" : "[result:value]"
}
}
]
}
}
You'll need to tweak the Rule to match your conf, but something like that should work.