There is a PDF that was turned into an Accordance tool (Hebrew Syntax Guide is what it's called in your library) which describes the syntax model. Now Robert Holmstedt who designed it is a semitics guy (sorry Robert for terrible gloss in summary) and the examples are all in Hebrew, but the principles apply to Greek also as the syntax module is based on the same underlying principles. I cannot recommend highly enough going through that document and it's examples pretty much cover to cover (well it's electronic and so there are no covers but you know what I mean ). It will get you straight on terminology and the basic plot. I tend also to think that having some understanding of generative approach to grammar helps but opinions vary.
So having said all that, objects as you traditionally think of them don't exist in this model. Complements do and at times they serve the function of what is called an object in traditional grammar. (Please leap in Robert, others, and correct me as required).
So what you can search for is genitive complements. Now whether that is entirely what you want for the particular inquiry at hand I don't know. Usually what I do is I have an example construction to hand and I look at how it is represented in the syntax chart and I construct a search that finds that structure. If it's something more general drawn from a grammar for example then I do something fairly simple and examine cases and see what interests me and I go from there refining things.
Something like the above will get you some examples to look at. You do need to study the diagrams and see then what interests you. In the Accordance Exchange I posted a collection of syntax queries based on Wallace's Greek Grammar Beyond the Basics. It is here : http://www.accordanc...ce_searches.zip. In particular it goes through many of his genitive cases. What it may help with is issues (identifying and in some cases getting over them) in getting from traditional grammar constructions to how they are represented in the Accordance charts.
If you have a specific example of a construction of interest let ping it back and I can take a look further.
Sola lingua bona est lingua mortua
ἡ μόνη ἀγαθὴ γλῶσσα γλῶσσα νεκρὰ ἐστιν
lišanu ēdēnitu damqitu lišanu mītu
"Du stammst vom Herrn Adam und der Herrin Eva ab", sagte Aslan. "Und das ist zugleich Ehre genug, um das Häupt des ärmsten Bettlers zu erheben, und genug, um die Schultern des größten Kaisers auf Erden zu beugen. Sei zufrieden." Aslan, Die Chroniken von Narnia, Prinz Kaspian von Narnia. CS Lewis. Übersetzt von Wolfgang Holbein und Christian Rendel.
Accordance Syntax Search For Wallace's Greek Grammar Beyond the Basics : https://github.com/4...WallaceInSyntax
Accordance Configurations :
Mac : 2009 27" iMac
Windows : MSI GE72 7RE Apache Pro laptop
Intel Core Duo Intel i7 Kabylake
Android : Samsung Note III 5.0, Samsung Tab S3 7.0 and Lenovo TAB4 8" 7.1