I will post an answer here to begin the discussion and hope others will contribute.
First, Matt 22:44 is not alone in quoting Ps 110:1. It is also quoted in Mark 12:36, Luke 20:42, Acts 2:34. In all cases the Greek is almost identical and reads:
Εἶπεν Κύριος τῷ Κυρίῳ μου ... (= said Lord to the Lord of me ...)
Notice that in both cases, we have the same word but in a different case namely, Κύριος (= Lord, nominative) and Κυρίῳ (= Lord, dative). Thus, based on the Greek text, an honest translation has no choice but to translate in the normal way something like, "The Lord said to my Lord ... ".
This means that when the inspired writers Matthew, Mark and Luke quoted Ps 110:1 they chose to translate the Tetragrammaton יהוה (YHWH) as Κύριος or "Lord"; they also translated אדֹנִ֗י (Adoni) as Κύριος or "Lord" also!!
This is not unique to these few passages. In fact, every time the inspired NT writers quote an OT passage containing the Tetragrammaton, יהוה (YHWH) it is almost always translated as Κύριος or "Lord", and occasionally by θεός (theos) = "God".
Here is a longer list of when this sort of thing happens - that is the when Tetragrammaton in the OT is translated by "Lord" (unless noted otherwise) in the NT. There are no exceptions to this rule.
- Matt 4:7 quotes Deut 6:16
- Matt 4:10 quotes Deut 6:13
- Matt 5:33 quotes Num 30:2 & Deut 23:21
- Matt 21:9 quotes Ps 118:26
- Matt 21:42 quotes Ps 118:22
- Matt 22:37 quotes Deut 6:4, 5 & 10:12
- Matt 22:44 quotes Ps 110:1
- Matt 23:39 quotes Ps 118:26
- Matt 27:9, 10 quotes Zech 11:12, 13
- Mark 1:3 quotes Isa 40:3
- Mark 11:9 quotes Ps 118:26
- Mark 12:10, 11 quotes Ps 118:23
- Mark 12:29 quotes Deut 6:4
- Mark 12:30 quotes Deut 6:5
- Mark 12:36 quotes Ps 110:1
- Luke 3:4-6 quotes Isa 40:3-5
- Luke 4:18 quotes Isa 61:1
- Luke 4:19 quotes Isa 61:2
- Luke 10:27 quotes Deut 6:5
- Luke 13:35 & 38 quotes Ps 118:26
- Luke 20:37 quotes Ex 6:3
- Luke 20:42 quotes Ps 110:1
- John 1:23 quotes Isa 40:3
- John 6:45 quotes Isa 54:13 but uses θεοῦ for היְהֹוָ rather than Κυρίου.
- John 12:13 quotes Ps 118:26
- John 12:38 quotes Isa 53:1
- Acts 2:20 quotes Joel 2:31
- Acts 2:21 quotes Joel 2:32
- Acts 2:25 quotes Ps 16:8
- Acts 2:34 quotes Ps 110:1
- Acts 3:22 quotes Deut 18:15
- Acts 4:26 quotes Ps 2:2
- Acts 7:31-34 quotes Ex 3:4-7
- Acts 7:48-50 quotes Isa 66:1, 2
- Acts 13:47 quotes Isa 49:5, 6
- Acts 15:17 quotes Amos 9:12
- Rom 4:3 quotes Gen 15:6 but uses θεῷ instead of κυρίῳ.
- Rom 4:8 quotes Ps 32;1, 2
- Rom 9:28 quotes Isa 10:22, 23
- Rom 9:29 quotes Isa 1:9
- Rom 10:13 quotes Joel 2:32
- Rom 10:16 quotes Isa 53:1
- Rom 11:3 quotes 1 Kings 19:14
- Rom 12:19 quotes Deut 32:35, 36
- Rom 14:11 quotes Isa 45:23
- Rom 15:11 quotes Ps 117:1
- 1 Cor 1:31 quotes Jer 9:24
- 1 Cor 2:16 quotes Isa 40:13
- 1 Cor 3:20 quotes Ps 94:11
- 1 Cor 10:9 quotes Num 21:5, 6
- 1 Cor 10:26 quotes Ps 24:1
- 2 Cor 6:18 quotes 2 Sam 7:8
- Eph 5:19 paraphrases Ps 30:4 & 92:1
- Heb 10:30 quotes Deut 32:35, 36
- 1 Peter 2:3 quotes Ps 34:8, 9
- 1 Peter 3:12 quotes Ps 34:15, 16
(I hope I have not made too many typos. Corrections welcome.) Thus, if the inspired NT writers thought it acceptable to translate the Tetragrammaton as the Greek equivalent of "Lord" then it is good enough for me.