How can I check user value null or not within mysql syntax?

You can use

    if(!empty($age)){
    //do sql operation
    }

You can also add constraints if you want only specific age groups. example:if you want age group between 18 and 40

    if ($_POST["age"] < 18 || $_POST["age"] > 40){
     //print error message
    }
    else{
     //do sql operation
    }

SELECT
    *
FROM
    `ident`
WHERE
    FirstName = '$first_name'
    && Age != ''
    && Age = $age
    && Age IS NOT NULL;

if ($age !== null) {
$result = mysqli_query("select * from ident where FirstName = '$first_name' Age = $age");
//etc..
}

You weren't very clear in your question, so I'll provide for both PHP & mySQL:

mySQL

Use the IS NOT NULL. Reference

SELECT * FROM ident 
    WHERE FirstName = '$first_name' 
    AND Age IS NOT NULL 
    AND Age = $age

PHP

Use empty() or isset(). Reference.

if(!empty($age)){
    //execute mySQL
}

You should check this on PHP page before querying the database.

<?php
if(!empty($age)) {
    $result = mysqli_query("select * from ident where FirstName = '$first_name' AND Age = '$age'");
}
?>

This should be done on query building side, you might want to throw an exception if certain values are not met as expected

PHP Use is_null() method or as said by @Huey

if(isset($age) and !is_null($age) and intval($age) > 0){
    $result = mysqli_query("SELECT * FROM ident WHERE FirstName = '$first_name' AND Age = $age");
}