Surely they would have tried to take what arguably would have been the jewel of the Hellenic world, a key religious city to the orthodox, not to mention a strategic point to hold. It seems like they would never have had a chance though and probably would have gone over as badly as if America dropped an atomic bomb on Tokyo in terms of the other side's reaction.
It will be good if you could pinpoint which era you are most interested in.
Immediately after the fall of Constantinople, there was worry that the Ottomans would continue to the heart of Christian Europe, leading to calls by Pius II to crusade. This had tremendous impact in the Mediterranean, what with Castille and Aragon and their dominions in the Mediterranean benefiting from the papal dispensation, but the call to crusade itself was ineffective.
Venice was an enthusiastic backer of the crusade, but by themselves they could not take on the Ottoman empire. The ended up having to go between accommodation to outright enmity. The loss of Candia in Crete in the 1660s forced them to agree to ally with Christian powers, culminating in the successful defense of Malta and then the naval ictory in Lepanto.
The nature of galley warfare is such that outposts are needed to support campaigns. This is where the campaign of the Christian alliance faltered: they could not form a cohesive follow-up to Lepanto. As a result, the Mediterranean was split between the Ottoman east and Christian west.
Between the stalemate in the Mediterranean, and the quagmire in SE Europe, the weakness of the Greeks meant there was no credible attempt at Constantinople.
How do you mean "Hellenic Wars"? The conquest of the Byzantine Empire was largely completed by the time 1453 rolled around. Many of the areas that weren't, such as Athens and the islands, were actually under the control of the Franks (western Europeans) at that time. A few areas that held on, such as Mystras and Trabzon, never came close to posing a real threat and were duly conquered within a couple decades.
That being said, jumping forward a few centuries the newly independent Greece absolutely had its eye on Constantinople. The first century or so of Greece's history was more or less defined by an an uncertainty in its borders, and this all came to a head in the immediate aftermath of WWI. The Treaty of Sevres saw the Allies attempt to carve out the Ottoman Empire into various ethnic sections unfavorable to the Turks. As part of this Greece was granted the western coast, which was seen as "Greek" by the western powers due to the composition of the cities (the countryside, however, was heavily Turkish, which spelled doom for the plans). The Greeks saw this as a chance to enact the "Megali Idea" (Great Idea), an irredentist plan in which Greece would control more or less all the lands around the Aegean--including Constantinople (soon to be Istanbul).
The Greek army enjoyed early success, but it pushed its luck and was later smashed by the revived Turkish army under Ataturk, and the western powers, despite their promises, had no appetite to enter into another war. The end result was that Greece and Turkey agreed on a "population swap" in which the Greeks would be deported from Turkey and Turks from Greece, a plan that made sense to nationalist power politics but resulted in incalculable human suffering.