We can compare the first element of the second array and find it's index in the first array and then iterate over it comparing successive elements for both to check if the Array2 is completely contained in Array1. Something like this should work:

String[] Array1 = { "the", "quick", "brown", "fox", "jumps",
                 "over", "the", "lazy", "dog", "in", "the",
                 "barn" };

String[] Array2 = { "in", "the", "barn", "next", "to",
                 "the", "chickens" };
                 
int index = Array.IndexOf(Array1, Array2[0]);
int subArrayFlag = 1;

if(Array2.Length + index > Array1.Length)
{
    Console.WriteLine("Array2 cannot be sub array of Array1!");
}
else
{
    for(int i =0;i<Array2.Length; i++)
    {
        if(Array.IndexOf(Array1, Array2[i]) >= 0)
        {
            if(Array2[i] == Array1[index + i])
                continue;
            else
                subArrayFlag = 0;
                break;
        }
        else
        {
            subArrayFlag = 0;
        }
    }
    if(subArrayFlag == 1)
    {
        Console.WriteLine("Array2 is subarray of Array1!");
    }
    else
    {
        Console.WriteLine("Array2 is not sub array of Array1!");
    }
}

Console.WriteLine("The first occurrence of partial Array2 is at index {0}.", index);

The above code can detect if the entire Array2 is contained in Array1 or not!

You can create a loop and use IndexOf until you got a result:

String[] Array1 = { "the", "quick", "brown", "fox", "jumps","over", "the", "lazy", "dog", "in", "the","barn" };

String[] Array2 = { "in", "the", "barn", "next", "to","the", "chickens" };
                 
int index = -1;
        
for (int j=0; j < Array2.Length; j++){
    index = Array.IndexOf(Array1, Array2[j]);
    if(index >= 0)
       break;
}

You can use the below method to get the first occurrence index in the Array1. In this way, you don't have to worry about values that exist in Array2 but not Array1.

private int GetFirstOccurenceIndex(string[] arr1, string[] arr2)
    {
        var keyIndexDict = new Dictionary<string, int>();

        for (var i = 0; i < arr1.Length; i++)
        {
            var key = arr1[i];
            if (!keyIndexDict.ContainsKey(key))
                keyIndexDict[key] = i;
        }

        foreach (var key in arr2)
        {
            if (keyIndexDict.TryGetValue(key, out var arr1Index))
                return arr1Index;
        }
        return -1;
    }

Example:

string[] arr1 =
{
        "the", "quick", "brown", "fox", "jumps",
        "over", "the", "lazy", "dog", "in", "the",
        "barn"
};

string[] arr2 =
{
       "someValueDoesntExistInArray1", "in", "the", "barn", "next", "to",
        "the", "chickens"
};


int index = GetFirstOccurenceIndex(arr1, arr2);
Console.WriteLine("The first occurrence of partial Array2 is at index {0}.", index);