This is just to add to Mike's answer, not to replace it.
Joshua does not transliterate into Greek exactly. There are letters in Hebrew that are simply not there in Greek. The Greek of Luke 3:29, Acts 7:45 and Hebrews 4:8 all have Ἰησοῦ/s for Joshua. Translators render it as Joshua instead of Jesus because that is the name readers will be familiar with. Likewise, the Septuagint (the Greek translation of the Old Testament often abbreviated LXX) uses Ἰησοῦ/s (Greek grammar rules specify that the final sigma appears depending on the case of the noun).
That "Joshua" transliterates as "Jesu"s is easy to see when you examine the respective alphabets (Hebrew, Greek).
J - the Hebrew yod becomes the Greek Iota
E - the same sound is found in both
H - Greek has no stand alone letter for H, so they had to drop this letter.
O - Without an H to connect to, the O disappears. Combining the e and o would produce an unnatural sound in Greek--they don't have that dipthong.
SH - the Hebrew shin (long e sound) is SH together and becomes a sigma (merely an S) as Greek does not have a letter for the SH sound.
U - equivalent sounds in both languages
A - Greek prefers not to end a name with a vowel sound, so they often (but not always) add a sigma.
The same differences with shin and a final vowel can be found in the Hebrew name Moshe, which we know better by the Greek Moses. You can also see such name changes in the Hebrew Shlomoh whom we know as Solomon. Greek does not have a equivalent for H so drops it, ending the name in a vowel (which they don't like), and adding an N.
Knowing these rules, and seeing how Joshua is rendered as Ἰησοῦ/s in the Septuagint and in the New Testament (and we know who it is where it is followed by "son of Nun") would be why scholars like Strong have linked them.