Essentially, yes. The typical "modern day mosque" you are thinking of is modeled off of classical Ottoman mosques, which grew up in Istanbul and were greatly inspired by Aya Sofya. The most important Ottoman mosque in the city, the Suleymaniye, was designed by its famous architect Sinan to exceed the church in dome diameter; the well known Blue Mosque was placed directly across from the Aya Sofya to serve as a sort of counterpoint to it.

Look at Gulru Necipoglu, Age of Sinan for a nice illustrated discussion of the motivations of Classical Ottoman architecture.

Edit: take a look at mosques from earlier periods or other regions in the Islamic world, and see the contrast. The mosques of Qayrawan, Cordoba, many Mamluk mosques in Cairo, the mosques of Samarqand, Isfahan, Delhi... Some, but not all of these have domes, but none look "typical" as you describe it

I'd say you're at least partially right. If you visit Istanbul today it's very easy to see that the architecture of Hagia Sophia had a very large impact on the construction of mosques in the city. There is a good reason why it is so often confused with the nearby Blue Mosque from the beginning of the 17th century (I've even seen a picture of the later being captioned as Hagia Sophia in Averil Cameron's The Byzantines).

First we should probably take a closer look to it's structural features. The building itself still gives you an unusually good impression of what it looked like in the 6th century. The Ottoman alterations to it are actually quite limited. From the outside there are as you mentioned the four minarets that were used for the islamic call to prayer and then there are some mausolea of the Ottoman sultans. On the other hand the great courtyard or atrium that originally preceded the church on it's western side is gone. On the inside there is the Mihrab niche inserted into the apse to signify the direction to Mecca, the Minbar, a sort of pulpit from where the Imam could deliver his sermons, and a lodge for the sultan. The chandeliers and the large disks featuring the names of Allah, Muhammad and the first four caliphs are also very prominent. Christian liturgical installations like the altar, the patriarch's throne or the ambo, from where the christian sermons were held, have obviously been removed. Many of the figural mosaics are also still covered up like they were for much of the Ottoman period. But this actually kind of restores the original look of the 6th century, as those mosaics all originate from later epochs of Byzantine history. On the hole I can't think of any late antique building that is this well preserved.

Like with any late antique church building the composition of the inside of Hagia Sophia was much more important than how it looked like from the outside. It was here where the christian congregation gathered, where the sermons where read and the holy Eucharist was celebrated. This contrasts with ancient temples, which were mostly used as a home for the statue of the god while the actual worship happened outside at the altar where the offerings were made. That's why ancient temple architecture didn't much influence the building of churches like the Hagia Sophia. They were just used in too different a way. The architects of Hagia Sophia did a fantastic job in creating an awe-inspiring stage for the christian liturgy. The interior is dominated by the large central dome. Unlike with similarly vaulted structures this dome is not immediately carried by massive walls, pillars or columns but instead balanced on a series of smaller half cupolas cascading down the east and west side, which makes it seem almost weightless. By adding flat walls on the north and south side the architects gave a longitudinal orientation to the building while still retaining a strong centralizing focus in form of the dome. This kind of combination of elongating and centralizing features was a hallmark of 6th century church building but nowhere was it achieved as harmoniously as at Hagia Sophia.

It's not hard to see why the Ottoman conquerors where impressed with this building and almost immediately began to adopt some of its features for their own building projects. The first large mosque to be build in Constantinople after the conquest of 1453 was that of sultan Mehmed II Fatih. It featured a large central dome carried by flat walls in the north, south and west and one large half cupola in the east. Like at Hagia Sophia lower side aisles were added at the north and south side of the central part of the building. Unfortunately the original building does not exist any more. The one that can be visited on site originates from the 18th century and doesn't preserve the original features. We are more lucky with the mosque of his successor Bayezid II which still exists. With two large half cupolas on the east and west side it comes even closer to the Byzantine model than the building of his father did. But the mosque that emulates Hagia Sophia most closely certainly is the one that Mimar Sinan build for sultan Suleiman the Magnificent from 1550 to 1558, almost exactly one thousand years after Justinian had set in motion his own building project. The similarities in its interior are immediately apparent. The central dome is followed further down by two flat walls in the north and south and the characteristic cascade of half cupolas in the east and west. The most prominent structural differences are probably the flat wall instead of an apse at the east end of the building and the prominent positioning of the four pillars supporting the dome. In the Hagia Sophia those pillars are much more concealed, again probably to add to the seeming weightlessness of the dome. Later Ottoman mosques would again stray a bit further from Hagia Sophia's model while still very much making use of many of its features. For example many would add another two half cupolas at the north and south side of the building, abandoning much of the longitudinal focus of the original. A famous example would be the aforementioned Blue Mosque.

So the impact Hagia Sophia had on Ottoman architecture is fairly obvious. But it is important to note that mosque architecture as a hole is far to diverse to feature only one type of building. When building a mosque architects can fall back on a huge pool of different traditions. For example look inside the national mosque of Qatar, the Imam Muhammad ibn Abd al-Wahhab Mosque in Doha, which opened only a few years ago. It does feature a hole host of domes but none dominates the whole building like it is the case in Hagia Sophia. Instead the focus seems to be on creating a virtual forest of columns. In this it is very similar to some of the earliest preserved mosque buildings like the Umayyad Mosque in Damascus (early 8th century) or the Mezquita in Cordoba (begun in second half of the the 8th century). I'm pretty sure that referring back to those ancient times was the intent of the rulers of Qatar.

If we go back to Ottoman architecture we can see that it is also much richer than just being a kind of remix of 6th century Byzantine prototypes. Ottoman architects had already been experimenting with domed mosques for some time before 1453. One of the most common types was simple domed room with square ground plan and a columned vestibule. The 14th century Yeşil Camii at Iznik is a nice example. Mosques utilizing this relatively simple plan are still being build all over Turkey. Large domes that dominated the interior of a building had probably entered Ottoman architecture by the way of Persia where they had been used in mosques at least since the 11th century.

Tl;dr: Yes, the Hagia Sophia very much had an impact on Ottoman mosque building but this is only one part of the much larger history of islamic architecture. Mosque architecture is as diverse as that of christian churches and there is much more than one type of mosque.

See: Howard Crane, Art and architecture, 1300-1453, in: Kate Fleet (ed.), The Camebridge history of Turkey. Vol. 1. Byzantium to Turkey, 1071-1453 (2009), pp. 266-352

Robert Hillenbrand, Islamic Art and Architecture (1998)

Doğan Kuban, Ottoman architecture (2010)