Cost-of-Quality: COQ is a relative term!

Components of the Cost-of-Quality in Kaizen parlance are as follows.

1/ Costs of external-failures incurred due to failures at customer's end.

2/ Costs of internal-failures incurred in order to handle failures at supplier's end.

3/ Costs of appraisal incurred in order to inspect/audit the adequacy and compliance of quality and quantity of work, and the workmanship as well, as per standards/procedures.

4/ Costs of prevention incurred in order to undertake improvement or failure-prevention activities (as preventative/proactive measures) & corresponding education/training for the same.

First two components are the 'cost-of-poor-quality' (CoPQ) that are incurred after failures. 

Last two components are the 'cost-of-good-quality' (CoGQ) or you can say 'investment' required to prevent a failures in the first place.

The components, rather it's terminology, is relative and hence the  term Cost-of-Quality itself is relative! Use of the components may depend upon the context of the situation.

Is the statement: "If the customer needs to administer incoming inspection on his inputs received from the supplier then that's typically the 'cost of appraisal'!", made by a participant in one of my sessions, 'true' or 'false', for instance?

The statement is 'false'!

It's actually the 'cost of prevention' for the customer. Because she wants to prevent defective inputs from going into the next process. Engaging into inspection activities on the customer's own processes or processing activities, although it's a NVA (non-value-activity or waste) however, is the 'cost of appraisal' for the customer in order to prevent the 'cost of internal failure' in the customer's processes.

Whereas the incoming inspection on his inputs may be is the 'cost of external failure' for the supplier. This is particularly so if the customer engages into 'inspection', a NVA (non-value-activity) 'forced' on to her by herself, because of lack of trust (past dissatisfaction) in the supplier's quality and hence in the supplier's brand image.

Also read a few relevant blogposts hereunder:

Do You Force Customers To Quit
In-a-problem ? No-problem! Dwell-a-while!
Customer or Custo-Mer ?  
Experienced A Delightful Payment ! 
An Experience of Heart-and-Soul 
Do you keep curing your brand ?  
A customer gets what s/he deserves: Shoddy Quality!  
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Should one care for value ?  
Can-changing-thoughts-change-a-nation
Ridiculous Poison-culture versus Maverick Kaizen-culture 
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Taken-For-Granted ? You Deserve It !! 
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Appraisal or Prevention! What's better?

In-process inspection inside a company is typically a 'cost of appraisal' for the company. 

Is the same true for a supplier, for her 'visit to customer for in-process inspection while installing the supplies' at customer's end as a part of the supplier's process!?

It's True, the said installation being a part of the process of the supplier company. 

It's, however, a 'cost of external failure' for the supplier if installation is customer's responsibility and she finds a defect/error in supplies while doing so. Otherwise, it's the 'cost of prevention' for the customer. 

In either of the cases, be it appraisal or prevention, it is a COGQ (cost of good quality) anyway as a part of the COQ (cost of quality)!

One may have to strike a balance though, while deciding upon spending on the latter, meaning on the visit to customer as a part of the process itself, especially in case of high-impact high-value industrial products. 

In any case, the COGQ in general and the other 'costs of prevention' in-house, in particular, may prove to be most beneficial and highly profitable at that.

Why not try validating this statement by calculating the components of Cost of  Quality, for a failed item that reached your customers in past!

Calculating the Cost of Quality (both the COPQ & COGQ), classifying the corresponding components as Tangible (Visible) Costs & Hidden Costs, and comparing them might help.

Also read a few relevant blogposts hereunder:

Do You Force Customers To Quit
In-a-problem ? No-problem! Dwell-a-while!
Customer or Custo-Mer ?  
Experienced A Delightful Payment ! 
An Experience of Heart-and-Soul 
Do you keep curing your brand ?  
A customer gets what s/he deserves: Shoddy Quality!  
Listen to iceberg of VOC to acquire customers 
Less With More And More Gets Sore 
Should one care for value ?  
Can-changing-thoughts-change-a-nation
Ridiculous Poison-culture versus Maverick Kaizen-culture 
Tolerate Once, Twice, Thrice?  
The Business Of Businesses 
That's how some business partnerships work 
Taken-For-Granted ? You Deserve It !! 
Strategise To Achieve Targets Daily 
Does recognition really matter 
That's How Morons Work
Simplicity Of Theory Of Relativity