Are the English surname "Douglas" and the Old Slavonic word глас (glas) "voice" cognates?

(I’m not sure this is really about English as such, given that one is a Celtic word and the other a Slavic word – but Douglas is an English name, so let’s give it the benefit of the doubt.)

tl;dr

These two words are not cognates, no.

• In the Slavic word, the reference to the voice as a stream of words is secondary: the primary meaning is the sound itself.
• In the Celtic word, the reference to streams of water is also secondary: the primary meaning is the blue-greenish colour of bodies of water.

In other words, they belong to two completely separate groups of words whose meanings happen to have been extended in such a way that they overlap somewhat.

Slavic глас

The Wiktionary article on Proto-Slavic *golsъ (the direct ancestor of the Old Church Slavonic word) derives the word from the Proto-Indo-European root *gels- meaning ‘call’, but this is not a commonly accepted root – I’m not sure where it was taken from here. A root *gelH- (or perhaps *gleH-) has been suggested by at least Peter Schrijver, but it is not included in LIV².

What is certain, however, is that there is a cluster of words relating to voice, calling, noise, etc., which have a structure that points back to a PIE consonant skeleton of *gl(H)(s); the Wiktionary article on call gives a good selection of cognates.

Importantly, none of these words refer to streams or anything like that – they all refer to vocal sounds. Some, like Latin gallus ‘rooster’, would be quite far-fetched if the original meaning of the root had been ‘stream [of words]’.

Goidelic glas

As you mention, Douglas comes from Scottish dubhghlas ‘blackstream’, which also exists in Irish and Manx, the other two languages in the Goidelic branch of the Celtic languages. In this word, the second part, glas, does indeed mean a stream, but that does not appear to be the original meaning – the original (and to this day more common) meaning of the word is ‘blue-green’ (can refer to either blue or green, depending on context). Cognates from continental Celtic languages like Gaulish show that this word comes from Proto-Celtic *glasto- with the same meaning: in Goidelic, *st became ss (reduced to s when word-initial or word-final), but the attested Gaulish form is glastum with the t intact, a derived noun that refers to woad.

Ranko Matasović’s Etymological Dictionary of Proto-Celtic takes PC *glasto- as being from PIE *ĝʰlh₃-stó-, a derivation from the PIE root *ĝʰelh₃- ‘gleam, glow; green, yellow’. This root underlies a whole slew of words related to green-, yellow- or shininess, from Greek χλωρός (whence chlorine, from its colour) to English yellow.

The meaning of the root changed a bit in Celtic: where in PIE, it referred to the part of the colour spectrum ranging from yellowish to greenish, in Celtic it had shifted to the part ranging from greenish to bluish. This includes the colour of various bodies of water which then metonymically became ‘the blue-green ones’ (cf. how phrases like ‘out of the blue’ refer to air as being blue, even though it’s not). Even in Modern Irish (perhaps Scottish too, I’m not sure), the ocean can still be called glasmhuir (‘blue-green sea’).

So Dubhghlas means ‘dark stream’, yes, but really it means ‘the dark blue-green one’.