'Aποφθέγγομαι, from Acts 2:4

The last phrase of Acts 2:4 is

καθὼς τὸ Πνεῦμα ἐδίδου ἀποφθέγγεσθαι αὐτοῖς

as the Spirit was giving to utter forth to them

Ἐδίδου is the 3-S, imperfect active indicative of the -μι verb δίδωμι: 'it was giving'.  By sense this verb can take an infinitive, and thus we have ἀποφθέγγεσθαι, the present middle/passive infinitive of ἀποφθέγγομαι, 'I speak out, I declare'.

it (the Spirit) was giving to them to speak out

I would call ἀποφθέγγεσθαι a deponent verb.  It is used three times in the New Testament, all of them in Acts.

Ἀποφθέγγομαι is a compound verb, with the preposition ἀπό preceding the root verb φθέγγομαι, 'I speak aloud, I utter'.  Φθέγγομαι itself is used only three times in the New Testament; once in Acts, and twice in 2 Peter.  

Note this use:

ὑποζύγιον ἄφωνον ἐν ἀνθρώπου φωνῇ φθεγξάμενον 

a speechless donkey spoke with human voice (2 Peter 2:16)

Although difficult to recognize - at least for me - the form φθεγζάμενον is, yes, a form of φθέγγομαι: an aorist middle (deponent) participle, neuter nominative singular.  Neuter because it refers to the donkey (ὑποζύγιον - i.e., 'under a yoke').

At first I was stumped as to any English relative of φθέγγομαι, but I then realized that a Greek relative of this verb is the noun φθόγγος, -ου, ὁ, 'a sound'.

And yes, the English 'diphthong' (literally, 'two sounds') comes from φθόγγος, although oddly - as best I can tell - 'diphtheria' comes from a completely different Greek word.