I am not an expert on Israeli-Azeri relations, but I can tell you that Azerbaijan has partly followed Turkey’s lead in foreign affairs. Turkey has traditionally had a strong relationship with Israel (particularly in terms of security cooperation) though this has changed in the last ten years. However, Israel treads very carefully with Turkey, especially on symbolic issues, refusing for instance to formally recognize the Armenian genocide.

Azerbaijan’s reorentation was fairly quick: Azeribaijan’s number one foreign concern is probably the contested region of Karabakh, and in the Karabakh conflict, Russia mostly supported the Armenians, whereas Azerbaijan sought help from Turkey. This started of course before Azerbaijan was even independent from the Soviet Union. But I think perhaps we should see Azerbaijan’s foreign policy as distinctly pro-Turkey, and Turkey is of course a long term NATO ally, whereas Armenian has generally had a pro-Russian foreign policy. While people often make too much of ethnicity, in this case these ties are longstanding: Russians saw themselves as protectors of the Christian minorities in the Ottoman Empire, including the Armenians, and Turkic ideology often talks about the unity of Turkic peoples, particularly Azeris and Turks. There is, of course, a dialect continuum of Turkish from Bulgaria to Baku, and natively the “Turks” and “Azeris” in northeast Turkey are mainly defined by religion (Sunni vs. Shia). There are a lot of monuments in both Azerbaijan and Turkey that proclaim “one people, two states” in Turkish and Azeri. I believe the Group of Russian Forces in the Transcaucasus almost immediately were kicked of Azerbaijan almost as soon as was possible. Since they were kicked out of Georgia in 2006, I believe Russian forces in the Transcaucus are only in Armenia and the breakaway areas of Georgia. Azerbaijan continues to see Armenia’s victory in the Karabakh war as largely due to military aid from Russia (as well as economic help from the Armenian diaspora). Russia’s continued close relations with Armenia obviously encouraged Azerbaijan to seek relations elsewhere, though it remains the chief weapons supplier of Azerbaijan’s Soviet legacy military.

Though they’re not domestically politically significant, it’s perhaps worth noting that Azerbaijan has one of the largest Jewish communities in the Middle East (smaller than Turkey’s but about equal with Iran’s).

The two have extensive security coordination. This is out of the 20-year window of the sub, but in the original Wikileaks dump of foreign service cables, one quotes Aliyev as saying that their relationship is like an iceberg with “nine-tenths of it is below the surface." Beyond Armenia, some of Azerbaijan’s most pressing concerns are shared with Israel, namely Islamist terrorism and an ambitious Iran.

We should also keep in mind that the Arab-Israeli conflict, on the level of foreign policy at least, is often not a broader Muslims-Israel conflict. In the oil embargo after the 1973 Yom Kippur War, all the Arab members of OPEC participated in the boycott, but, for example, Muslim-majority OPEC member Indonesia did not.