You don't need to be born with perfect pitch to hear music notes. Learning to hear notes and recognize the names of those notes takes time and practice. It may be that you never develop perfect pitch, which is the ability to hear a note and name it, but you can develop relative pitch: the ability to recognize musical intervals, match tones so you can play songs by ear and recognize notes within the context of a scale or song.
Use a musical instrument as a reference. A piano is the preferred instrument, but a guitar will do as well. Be sure the instrument is in tune. Start with a C scale and play it slowly, in ascending and descending order. The notes of the C major scale are C, D, E, F, G, A, B, C. This is a full octave. There are no sharps or flats. Play the notes with your eyes open a few times, concentrating on the sound of each. Now play the notes with your eyes closed so you can't see your fingers touching the keys. Concentrate on each note and listen for the very slight difference in the tone of each. In particular, pay attention to the first and eighth notes of the scale. Though they are an octave apart, let your ears hear how the color of the tone is the same for these two notes. Practice this exercise with all 12 major keys.
Play broken chords and listen to the intervals. An interval is the distance between two notes. A major chord is made of of the first, third and fifth notes of the major scale of the same name. A C major chord will be made up of the first, third and fifth notes of the C major scale you played earlier. Those notes are C, E and G. Play those three notes in the C major scale you played earlier. Focus on listening to the intervals between the first note and the third. When you get used to that note, play the third note flatted. This is the note that turns a major chord into a minor chord. In a C chord, you would play an E flat rather than an E. Now play the all of the notes together to make a C major chord, then immediately flat the third note to make a C minor. Pay attention to the difference that single flatted note makes between a major chord and a minor. These exercises will help your ear distinguish intervals and tones, which is the basis for learning to hear notes.
Locate the C major scale in two positions on your instrument. Play the first note, C, in one of the positions, then close your eyes and find the same note in the second position scale. Continue playing notes in the second position scale until you match the note you played in the first.
Practice these listening techniques in every major and minor scale. Try associating a color to each note of the scale. When you play the scales back, select one of your colors and find the note you associate with that color. By assigning colors to different notes, you add a visual reference that enhances your auditory senses. If you assign the color blue to the C note, then assign a lighter shade of blue to a C flat and a darker shade of blue to a C sharp.
Pick a note you want to make and hum it. When you feel you have the note you wanted to hum, play the note on your instrument and see if you are humming the correct note. If you aren't, then raise or lower your hum until you hear yourself come into tune with the note you're playing. With practice, you'll be able to call off any note and hum or sing it correctly.
Try Solfege. It's free ear training software that can help you develop the ability to train yourself to hear and recognize musical notes (see Resources).