[SalesForce] How to get the value of an Array of Objects present inside an Array of objects

You can use ActionFunction for this. Here is sample for your reference.

Controller:

public class ActionFunctionCLS {

    public void IWantToDebug() {
        list<String> ls = ( List<String> )JSON.deserialize( Apexpages.currentPage().getParameters().get('node'), List<string>.class ) ;        
        System.debug('======================= ' + ls[0] );
        System.debug('======================= ' + ls[1]);
        System.debug('======================= ' + ls[2]);                
    }
}

VF Page:

    <apex:page controller="ActionFunctionCLS"  >

<script type="text/javascript">
    function doSave(node){
        var arr = ["foo", "bar", "baz"];
        paraFunction(JSON.stringify(arr));
    }       
</script>

<apex:form >
    <apex:actionFunction name="paraFunction" action="{!IWantToDebug}" rerender="view">      
         <apex:param id="anode" name="node" value="" />
   </apex:actionFunction> 
</apex:form>

 <input type="button" onclick="doSave();" value="test" />

Note that we are calling IWantToDebug Action from JS using paraFunction Action Function.

My advice is to keep the query in Javascript. Much simpler to keep all the logic in one place than try to route server logic into client code when you can avoid it.

You can, however, use a serialized array property.

public String mySerializedList { get; private set; }
public MyConstructor()
{
    mySerializedList = JSON.serialize([/*some query*/]);
}

Then in your client-side application you can parse it.

var objectArray = JSON.parse('{!JSINHTMLENCODE(mySerializedList)}');