According to Murray J. Harris, the reader must assume the writer of Hebrews did not mean ὁ θεός ("God"):
When (ὁ) θεός is used, we are to assume that the NT writers have in mind ὁ πατήρ unless the context makes this sense of (ὁ) θεός impossible.
If this were so the letter is meant to be begin as:
God [The Father], having spoken long-ago in-many-portions and in-many-ways to the fathers by the prophets, spoke to us at the last of these days by a Son, Whom He appointed inheritor of all things, through Whom also He made the worlds (Hebrews 1:1-2 DLNT)
πολυμερῶς καὶ πολυτρόπως πάλαι ὁ θεὸς [ὁ πατήρ] λαλήσας τοῖς πατράσιν ἐν τοῖς προφήταις ἐπ᾽ ἐσχάτου τῶν ἡμερῶν τούτων ἐλάλησεν ἡμῖν ἐν υἱῷ ὃν ἔθηκεν κληρονόμον πάντων δι᾽ οὗ καὶ ἐποίησεν τοὺς αἰῶνας
Applying Harris' conclusion to Hebrews is speculative. "God the Father," is never used in this letter so there is no internal evidence in the letter to support this position. As the letter is anonymous, there is no definitive support from elsewhere in the NT. There is also a question why the writer would use only υἱῷ, not υἱῷ αὐτοῦ which must be addressed before presuming ὁ πατήρ is implied. Also, there is the practical issue of presuming a letter written to prevent someone from abandoning sound doctrine would begin by writing something other than they meant.
"The Father" in the OT
The New Testament, "God the Father," is applied differently in the Old Testament:
For you are our Father, though Abraham does not know us, and Israel does not acknowledge us; you, O LORD, are our Father, our Redeemer from of old is your name. (Isaiah 63:16) [ESV]
The verse reads: For thou art our Father; for Abraham knoweth us not and Israel doth not recognise us; Thou Jehovah art our Father; our Redeemer from of old is Thy Name. Jehovah is the Father of Israel, i.e. the Creator and founder of the nation (Deuteronomy 32:6; Malachi 2:10; cf. Exodus 4:22; Hosea 11:1; Isaiah 1:2; Jeremiah 3:4; Jeremiah 3:19; Malachi 1:6). The idea of the divine Fatherhood is not yet extended in the O.T. to the individual believer, although a remarkable anticipation of the N.T. doctrine is found in Sir 23:1; Sir 23:4 : “O Lord, Father and Master of my life, … O Lord, Father and God of my life.” (Cheyne.)
One might consider "God as Father" in the context of the nation of Israel (cf. Exodus 4:22-23), but not with "Son" as in the letter.
When looking to the Old Testament, ὁ θεὸς would more reasonably be seen as referring to אלהים (Elohim) not יהוה (YHVH). This may be seen in the letter:
8 But of the Son he says, “Your throne, O God, is forever and ever, the scepter of uprightness is the scepter of your kingdom. 9 You have loved righteousness and hated wickedness; therefore God, your God, has anointed you with the oil of gladness beyond your companions.” (1:8)
8 πρὸς δὲ τὸν υἱόν ὁ θρόνος σου ὁ θεός εἰς τὸν αἰῶνα τοῦ αἰῶνος καὶ ἡ ῥάβδος τῆς εὐθύτητος ῥάβδος τῆς βασιλείας σου 9 ἠγάπησας δικαιοσύνην καὶ ἐμίσησας ἀνομίαν διὰ τοῦτο ἔχρισέν σε ὁ θεός ὁ θεός σου ἔλαιον ἀγαλλιάσεως παρὰ τοὺς μετόχους σου
6 Your throne, O God, is forever and ever. The scepter of your kingdom is a scepter of uprightness; 7 you have loved righteousness and hated wickedness. Therefore God, your God, has anointed you with the oil of gladness beyond your companions (Psalm 45)
כִּסְאֲךָ אֱלֹהִים עֹולָם וָעֶד שֵׁבֶט מִישֹׁר שֵׁבֶט מַלְכוּתֶֽךָ׃ 6
אָהַבְתָּ צֶּדֶק וַתִּשְׂנָא רֶשַׁע עַל־כֵּן מְשָׁחֲךָ אֱלֹהִים אֱלֹהֶיךָ שֶׁמֶן שָׂשֹׂון מֵֽחֲבֵרֶֽיךָ׃7
ὁ θεός is consistently used for אֱלֹהִים.
Hebrews 1:1-2 says nothing about YHVH in the Old Testament.
1. Murray J. Harris, Jesus as God, WIPF & Stock Publishers, 1992, p. 47
2. Cambridge Bible Commentary