First, some background information:
In both the Torah She-Bi-Kethav and the Torah She-B`Al Peh, there is a concept of the Creator setting an example of proper ethical and moral conduct for mankind in general and for the Jewish Nation in particular. This concept of the imitation of God was developed further through the midrashic fervor of the Tanna'im and Amora'im in the Mishnah, Talmudhim, Agadoth, and other texts of Haza"l.
However, since logic tells us that the Creator does not possess positive attributes and is not subject to "accidents" (cf. Ramba"m Hilkhoth Yesodhei HaTorah 1:1ff; Pirush HaMishnayoth, Sanhedrin 10, and Moreh HaNevukhim I), ALL of these attributions must be seen as metaphorical and for our benefit as He, yith'aleh wa-yithborakh, is a Bilti Ba`al Zequq (One Who needs nothing).
The "examples" which have been [metaphorically] set for us by HaShem are in two categories:  how we must act one to another (ben adham le-havero), and  how we must act toward Him (ben adham le-Maqom).
The former are shown in the well-known examples of HaShem "performing" various misswoth, such as:
- Biqur Holim - Visiting The Sick - when He "visited" Avraham Avinu on the third day after his berith milah (cf. Rash"i, Parashath Wayera; b.Sotah 14a)
- Qevura - Burying The Dead - when He buried Mosheh Rabbenu (cf. b.Sotah 14a)
- Hatarath Nedarim - Annulment Of Vows - when He removed the gezerah from Yekhonyah HaMelekh after he did teshuvah in Bavel, which resulted in the conception of Shealti'el the father of Zerubavel (cf. b.Sanhedrin 38a)
The later, however, are not shown to us by how HaShem serves the Creator (has wa-shalom), as if such a thing were even possible (halilah wa-has), but rather - ironically - how HaShem serves - as it were - the Jewish Nation. That is, since all of His dealings with His creations are based on the concept of berith (an agreement between 2 or more parties), through seeing how HaShem treats His portion of the berith with `Am Yisrael we gain insight into how we should approach our duties as spelled out in our portion of the berith. In this vein, we find the following examples:
- Tefillin - In b.Berakhoth 6a, Haza"l discuss the concept of HaShem's "tefillin" and what passages they might contain. The conclusion is that the passages of His "tefillin" metaphorically correspond and are reciprocal to ours. Whereas ours contain (among others) the passage Shema` Yisra'el (Devarim 6:4) - expressing our unique relationship to HaShem - His contain the passage Mi KeAmekha Yisra'el (Divrei HaYamim 17:21)
- Limudh Torah - In various places (b.Avodah Zarah 3b; Yalqut Shim`oni, Parashath KiTisa "Wayeshev HaAm"; Tanna DeBe Eliyahu Rabbah, Pereq 13) the idea of HaShem studying Torah is mentioned to illustrate His steadfast devotion to His portion of the berith, i.e. His duties (as defined by the Torah) toward the Jewish Nation.
And it is this final example which leads to answering your question.
Now, to answer your question:
I would like to suggest, based on the above explanation and the simple wording of the Gemara in b.Avodah Zarah 3b, that the proper understanding of the passage is as follows:
The Gemara list four activities:
- Studying Torah
- Judging The World
- Providing Sustenance To All Creatures
- Playing with the Lewiyathan
The common thread between the later three (i.e. judging the world, providing sustenance for the world, and playing with the Lewiyathan) is that He is focused on tending to His creations. So then, it seems to stand to reason that HaShem's Torah study is intended to have a similar thrust. Thus, it would seem that the message for Jews in HaShem's studying of the Torah is "Just as the Creator is careful to constantly keep His duties toward the Jewish Nation in His heart and mind - as it were - so also we should dedicate a significant portion of our time (e.g. a quarter of the day) studying the Torah so that we can dutifully fulfill our portion of the berith with HaShem."
According to the Shulhan `Arukh (OH 1:1), it was a practice of the Ssadiqim to constantly think of HaShem and His greatness in accordance with the pasuq (Tehillim 16:8), Shiwithi HaShem leneghdi thamidh - "I have set HaShem before me always." I suspect that this was due - at least in part - to their understanding of the above concept.
Hope this was adequate.