I know that Muslims refer to God as "Allah" but do they recognize the tetragrammaton as at least one of God's names? IE, would it be wrong for a Muslim to refer to God by the tetragrammaton?
It is not permissible for a Muslim to call Allah by any other name or names except for those which have been stated in the QURAN and Hadith.
To simply take a note of this fact see what Allah ta'ala has revealed in this verse.
هُوَ اللَّهُ الْخَالِقُ الْبَارِئُ الْمُصَوِّرُ لَهُ الْأَسْمَاءُ الْحُسْنَى يُسَبِّحُ لَهُ مَا فِي السَّمَاوَاتِ وَالْأَرْضِ وَهُوَ الْعَزِيزُ الْحَكِيمُ
He is Allah , the Creator, the Inventor, the Fashioner; to Him belong the best names. Whatever is in the heavens and earth is exalting Him. And He is the Exalted in Might, the Wise.[59:24]
In another verse Allah subhanahu wa ta'ala mentions:
اللَّهُ لَا إِلَهَ إِلَّا هُوَ لَهُ الْأَسْمَاءُ الْحُسْنَى
Allah - there is no deity except Him. To Him belong the best names. [20:8]
And here in this ayah you can see the ORDER form Allah subhanahu wa ta'ala:
وَلِلَّهِ الْأَسْمَاءُ الْحُسْنَى فَادْعُوهُ بِهَا وَذَرُوا الَّذِينَ يُلْحِدُونَ فِي أَسْمَائِهِ سَيُجْزَوْنَ مَا كَانُوا يَعْمَلُونَ
And to Allah belong the best names, so invoke Him by them. And leave [the company of] those who practice deviation concerning His names. They will be recompensed for what they have been doing. [7:180]
Here Allah has ordered us to call Him by the names He have mentioned. And has prohibited us from deviating. Not only that but also those who practice this deviation, they will have to compensate for this act. Because you can call Allah only by the names that Allah has ordered us to.
Those who have claimed other names for Allah have no authority and base-root for doing so. To make it understandable, ...can we call anyone by any name that we wish to? Of course not. Because the person we are calling to has his own identity and calling anyone by other names rather than the names he likes or permits
(names which attribute to his/er virtues, family etc.), rather hurt him. Even this changing is also prohibited in the Quran i.e. changing any individuals name to any name that s/he does not like (
is prohibited in the QURAN).
I can't remember the exact verse. I'd be very glad if someone had edited my answer and provided that ayah.
Yahweh isn't mentioned exactly in the Quran or Hadith, and so using this name in this form isn't recommended since we can't be sure if its an innovation or really from God. The Jews themselves exercise caution and prefer to not use the tetragrammaton because of its sanctity and the prohibition of taking God's name in vain. They have used the name Eloah, Elohim and Elah (cognates of Allah) as well as Adoni etc.
With that said, the hebrew YHWH is semantically related to one of God's name in the Quran "Al-Haee" (The Ever-Living), and can be rendered as Ya-Haeeu. Al Haee is mentioned in for example Quran 2:255 and Quran 3:2
יְהוָ֔ה in Hebrew means the numbers 10 5 6 10 or letters. وَإِلَـهُكُمْ إِلَهٌ وَاحِدٌ لاَّ إِلَهَ إِلاَّ هُوَ الرَّحْمَنُ الرَّحِيمُ in this verse from Quran we have هُوَ which refers to Allah the first letter and second letter means 5 and 6 accordingly, which is interesting fact. Probably interesting facts can be found. This page can explain more and the letter Y just means "O" that appears before the word "Israel" in the biblical verse displayed above: " O Israel: Yahweh is our God", so the letter "Ya" is not actually part of God's name.
The name which means The Giver of Life in the Quran is Al-Muhyi. This name is mentioned 4 times in the Quran according to Wikipedia. If you go to these verses you will find Ya-Haeeyu ( يَا حَىُّ ), Yuhyee ( يُحيي ) words. Which resembles in pronunciation the letters YHWH I guess.
Note this one thing before anyone would question about God in Islam, we only have one God, that is Allah. As stated in the Quran (Al-Ikhlas, QS 112) https://quran.com/112 You can read it and actually the verses of Al-Ikhlas explained everything about our belief in one God. Maybe in other book of other religion called our God with a different name.
Other than that, as a Muslim scholar, I personally have never heard of that name before.
Name of God is YHWH. "Allah" means "The God". "Illah" means "God". These are just language translations like "Khuda". They can mean any of the Gods worshipped by Humans.