The 'Theological Dictionary of the N.T.' page 607 defines "monogenes" as only begotten."
I think here we do not need to expand much on "theos" as it means "god" or God" according to context etc..
Thus from the Greek "0nly-begoteen god"
Example of the meaning in use:-
'The N.T.', A new translation by Richmond Lattimore, "among the most distinguished translators of the Greek classics." John 1:18 "No one has ever seen God; the only-born God who is in the bosom of his father, it is he who told of him."
English
"begotten [bih-got-n] verb a past participle of beget.
beget [bih-get] verb (used with object), be·got or (Archaic) be·gat; be·got·ten or be·got; be·get·ting. (especially of a male parent) to procreate or generate (offspring). cause; produce as an effect: a belief that power begets power."-https://www.dictionary.com/browse/begotten
ADDED:
John 1:18 - “Only-begotten god”, others read “Only-begotten son.”
(ASV) No man hath seen God at any time; the only begotten Son, who is in the bosom of the Father, he hath declared him.
(Bishops) No man hath seene God at any tyme: The onely begotten sonne which is in the bosome of the father, he hath declared hym.
(Darby) No one has seen God at any time; the only-begotten Son, who is in the bosom of the Father, he hath declared him .
(DRB) No man hath seen God at any time: the only begotten Son who is in the Bosom of the Father, he hath declared him.
(EMTV) No one has seen God at any time. The only begotten Son, He who is in the bosom of the Father, He has declared Him.
(Geneva) No man hath seene God at any time: that onely begotten Sonne, which is in the bosome of the Father, he hath declared him.
(KJV) No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.
J. Strong’s Greek Dictionary No. 3439 μονογενής monogenēs mon-og-en-ace From G3441 and G1096; only born, that is, sole: - only (begotten, child).
(LITV) No one has seen God at any time; the only begotten Son, who is in the bosom of the Father, that One declares Him.
(MKJV) No one has seen God at any time; the Only-begotten Son, who is in the bosom of the Father, He has declared Him.
(Murdock) No man hath ever seen God; the only begotten God, he who is in the bosom of his Father, he hath declared [him].
(Webster) No man hath seen God at any time; the only begotten Son, who is in the bosom of the Father, he hath declared him.
(YLT) God no one hath ever seen; the only begotten Son, who is on the bosom of the Father--he did declare.
“ονογενης, ες, (μονος, γενW)Ep. and Ion. only-begotten."-‘The Greek-English Lexicon’ of H.G. Liddell, M.A., and R. Scott, M.A. page 475